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The Problem Is the Problem: Towards Scalable Mathematical Discovery
Zeyu Zheng, Shengtong Zhang, Jeremy Avigad, Prasad Tetali, Sean Welleck
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 92%
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Summary
The paper introduces Find, Attempt, and Recommend (FAR), a human-AI discovery paradigm for scalable mathematical research. FAR automates the identification of open problems from a literature corpus based on a specified research direction, attempts to resolve them using frontier AI models, and filters results for expert review. In a combinatorics pilot, the system processed over 5,000 papers to identify and resolve several conjectures, demonstrating an efficient allocation of scarce AI reasoning and human review resources.
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Relation Signals (11)
Zeyu Zheng → affiliatedwith → Carnegie Mellon University
confidence 95% · Zeyu Zheng ... Affiliation: Carnegie Mellon University
Shengtong Zhang → affiliatedwith → Anysphere Co.
confidence 95% · Shengtong Zhang ... Affiliation: Anysphere Co.
FAR → consistsof → Find, Attempt, and Recommend
confidence 95% · we build Find, Attempt, and Recommend (FAR), a literature-to-review cascade
FAR → proposesparadigmfor → scalable mathematical discovery
confidence 95% · We address them by proposing a new human-AI discovery paradigm... Towards Scalable Mathematical Discovery
FAR → discoversresultsfor → Lund–Saraf–Wolf
confidence 90% · Among them, we identify many interesting discoveries, including results on conjectures and questions of ... Lund--Saraf--Wolf
FAR → discoversresultsfor → Davies–Jenssen–Perkins–Roberts
confidence 90% · Among them, we identify many interesting discoveries, including results on conjectures and questions of Davies--Jenssen--Perkins--Roberts
FAR → discoversresultsfor → Erdős–Straus
confidence 90% · Among them, we identify many interesting discoveries, including results on conjectures and questions of ... Erdős--Straus
FAR → discoversresultsfor →
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Abstract
Abstract:AI systems are increasingly capable of contributing to mathematical research. In research practice, frontier-model reasoning is a limited resource, and expert mathematical review is even more sharply constrained. Allocating these scarce resources well is therefore central to making AI-assisted mathematical discovery efficient. In most current AI-for-math workflows, human effort is concentrated at the beginning and end, in selecting suitable research problems and later reviewing the resulting artifacts. These two stages are becoming bottlenecks for research-level mathematics. We address them by proposing a new human-AI discovery paradigm. The human input is no longer a single problem selected in advance, but a research direction in which the experts have interest and expertise. The system then searches a broad literature corpus for candidate problems in that direction. Inspired by search and recommender systems, we build Find, Attempt, and Recommend (FAR), a literature-to-review cascade that automates the search for suitable problems and focuses human attention on artifacts that have passed several stages of filtering. In a combinatorics pilot, the pipeline starts from 5,245 combinatorics papers, recovers 6,453 candidate conjectures or open problems, and filters them to 4,717 apparently well-posed and still-open conjectures. Subsequent reasoning and automated triage stages surface 598 potential resolutions and select 77 items for author-team review. Among them, we identify many interesting discoveries, including results on conjectures and questions of Davies--Jenssen--Perkins--Roberts, Erdős--Straus, Ikenmeyer--Pak--Panova, and Lund--Saraf--Wolf. These results demonstrate the effectiveness of this new mode of human-AI collaboration for mathematical discovery.
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- Source: https://arxiv.org/abs/2608.16977v1
- Canonical: https://arxiv.org/abs/2608.16977v1
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The Problem Is the Problem: Towards Scalable Mathematical Discovery Zeyu Zheng Thanks: Equal contribution. Affiliation: Carnegie Mellon University Shengtong Zhang11footnotemark: 1 Affiliation: Anysphere Co.zeyuzhen, avigad, ptetali, swelleck@andrew.cmu.edu,shengtong@anysphere.cohttps://github.com/zeyu-zheng/FAR Jeremy Avigad Affiliation: Carnegie Mellon University Prasad Tetali Affiliation: Carnegie Mellon University Sean Welleck Affiliation: Carnegie Mellon University Abstract AI systems are increasingly capable of contributing to mathematical research. In research practice, frontier-model reasoning is a limited resource, and expert mathematical review is even more sharply constrained. Allocating these scarce resources well is therefore central to making AI-assisted mathematical discovery efficient. In most current AI-for-math workflows, human effort is concentrated at the beginning and end, in selecting suitable research problems and later reviewing the resulting artifacts. These two stages are becoming bottlenecks for research-level mathematics. We address them by proposing a new human-AI discovery paradigm. The human input is no longer a single problem selected in advance, but a research direction in which the experts have interest and expertise. The system then searches a broad literature corpus for candidate problems in that direction. Inspired by search and recommender systems, we build Find, Attempt, and Recommend (FAR), a literature-to-review cascade that automates the search for suitable problems and focuses human attention on artifacts that have passed several stages of filtering. In a combinatorics pilot, the pipeline starts from 5,245 combinatorics papers, recovers 6,453 candidate conjectures or open problems, and filters them to 4,717 apparently well-posed and still-open conjectures. Subsequent reasoning and automated triage stages surface 598 potential resolutions11 1 Available at https://probxiv.com. and select 77 items for author-team review. Among them, we identify many interesting discoveries, including results on conjectures and questions of Davies–Jenssen–Perkins–Roberts, Erdős–Straus, Ikenmeyer–Pak–Panova, and Lund–Saraf–Wolf. These results demonstrate the effectiveness of this new mode of human-AI collaboration for mathematical discovery. 1 Introduction AI systems are increasingly capable of contributing to mathematical research, with recent progress on mathematical reasoning benchmarks (Hendrycks et al. 2021; Zheng et al. 2021; Guo et al. 2025; Shao et al. 2025), formal theorem proving (Trinh et al. 2024; Chervonyi et al. 2025; Hubert et al. 2026; Ren et al. 2025; Xin et al. 2025; Chen et al. 2025; Seed 2026), and selected research-level problems (OpenAI 2026; Alon et al. 2026; Tsoukalas et al. 2026; Team 2026). Most of these systems operate with a problem-level interface, in which a researcher supplies a theorem, conjecture, or formal goal, the system attempts it, and the output is checked. This interface is useful, but it leaves out an essential part of research: deciding which problems are worth attempting in the first place. We study the decision of which problems to attempt in terms of effort allocation for AI-assisted mathematical discovery. Frontier-model reasoning and expert mathematical review are scarce, and their value depends on which conjectures receive them. Rather than concentrating reasoning and review effort on a few problems chosen in advance, we propose a new workflow in which experts specify a research direction, and the AI system automatically finds problems to focus effort on (Figure 1). We build Find, Attempt, and Recommend (FAR), a literature-to-review cascade. Given a broad mathematical topic, FAR finds relevant open conjectures from a large literature corpus, attempts to prove or disprove them, and recommends promising conjecture-resolution pairs for expert review. Figure 1: From choosing a problem to choosing a direction. The upper part shows the problem-level interface; the lower part shows our approach. Figure 3 gives the details of FAR. We instantiate the workflow in combinatorics. Starting from 51,110 mathematics papers, the pipeline identifies 5,245 combinatorics papers, extracts 6,453 candidate conjectures or open problems from 2,742 papers, and filters these to 4,717 apparently well-posed and still-open conjectures. A first broad attempt run covers all 4,717 of them. Automated triage surfaces 598 potential resolutions, and a final selection step chooses 77 items for internal author-team review. We manually checked 15 of the resulting artifacts, chosen by our own interest. They include counterexamples to conjectures of Davies–Jenssen–Perkins–Roberts and of Lund–Saraf–Wolf, a proof of Ikenmeyer–Pak–Panova’s conjecture on symmetric-group characters, and an answer to a question of Erdős–Straus on divisibility among binomial coefficients. We study various strategies for allocating a budget of solving attempts in our pipeline. We frame allocation as a constrained optimization problem, and derive strategies for maximizing notions of quality and importance. We show that these strategies can lead to more successful artifacts than with a uniform baseline. Furthermore, the optimal strategy depends on the objective: for instance, the strategy differs if we wish to maximize the number of successful artifacts versus maximizing the importance across all artifacts. In summary, our contributions are as follows: • We formulate scalable AI-assisted mathematical discovery as an effort-allocation problem over a pool of interesting mathematical problems, leading to a human–AI collaboration paradigm for using frontier-model reasoning and expert mathematical review efficiently. • Inspired by search and recommender systems, we introduce Find, Attempt, and Recommend (FAR), which builds this problem pool from the mathematical literature and turns model attempts into reviewable artifacts. • We instantiate the workflow in a combinatorics pilot and analyze the resulting literature-to-review funnel, including strategies for allocating model attempts. • We obtain author-reviewed solutions to problems from the combinatorics literature, spanning proofs, counterexamples, and answers to open-ended questions. The write-ups are collected in Appendix C. 2 Motivation and Related Work 2.1 AI for Mathematical Research Recent AI systems have made rapid progress on the part of mathematical research that begins after a problem or objective has been specified. For example, FunSearch and AlphaEvolve search for new mathematical constructions, programs, and algorithms (Romera-Paredes et al. 2024; Novikov et al. 2025). AlphaProof Nexus studies formal proof search on open research-level problems (Tsoukalas et al. 2026). Aletheia, Rethlas, QED, and other recent pipelines explore autonomous or semi-autonomous workflows for attempting open mathematical problems (Feng et al. 2026; Ju et al. 2026; An et al. 2026; Ju et al. 2026; Peng et al. 2026). Frontier models have also contributed to individually selected problems of long-standing research interest, including OpenAI’s disproof of the Unit Distance Problem and the Fable-assisted counterexample to the Jacobian conjecture (OpenAI 2026; Alon et al. 2026; Alpöge 2026; Bukh et al. 2025). Related work on an AI co-mathematician (Zheng et al. 2026) develops a collaborative framework in which AI agents pursue parallel workstreams while mathematicians steer the research process. Taken together, these works point toward what Tao describes as a transition from proof scarcity to proof abundance (Tao 2026). Mathematical research, however, rarely starts from an isolated problem statement. Before trying to solve a problem, researchers often need to study a direction of interest, find problems that matter within it, and decide whether those problems are worth sustained thought. These steps are ordinary parts of mathematical work, but they mostly sit outside the scope of today’s AI-for-math systems. In current uses of AI for mathematics, the human work before and after model reasoning—problem selection and expert review—is becoming the narrow part of the pipeline. Our approach moves the starting point of AI assistance to this earlier stage of mathematical research. Rather than selecting a problem for the system, mathematicians specify a research direction. The system searches an available literature corpus, recovers and attempts candidate problems in that direction at scale, and returns a small set of artifacts for mathematical review. This shift is analogous to the move from task-driven agents, which act on a specified task, toward more proactive systems that help surface what tasks are worth pursuing (Zhou et al. 2026; Lu et al. 2025; Lab 2026). The human-specified direction bounds the system’s initiative and aligns the search with the mathematicians’ interests and domain expertise. Mathematicians remain responsible for validating and reporting any resulting mathematical claims (Alper et al. 2026; Shan et al. 2026). 2.2 Constructing an Attemptable Pool A research direction does not by itself provide the system with a set of attemptable problems. Mathematical benchmarks such as PutnamBench, FrontierMath, and FirstProof provide clean, self-contained problem pools (Tsoukalas et al. 2024; Glazer et al. 2024; Abouzaid et al. 2026). In software engineering, another major domain for LLM agents (Yang et al. 2024), GitHub provides a centralized and structured collection of repositories, issues, documentation, and executable software from which benchmarks such as SWE-bench and ProgramBench can be constructed (Jimenez et al. 2024; Yang et al. 2026). Mathematics does have valuable collections, including the Open Problem Garden, AIM Problem Lists, and Formal Conjectures (Open Problem Garden 2026; American Institute of Mathematics 2026; Firsching et al. 2026). Their coverage is selective, however, and they are not the primary infrastructure in which mathematical problems and their surrounding research context are recorded. Much of this information remains dispersed across the literature. A conjecture may appear as a numbered statement, a question, a remark, an unresolved case, or a sentence embedded in local notation, and its status may change after publication. Constructing an attemptable pool of problems therefore requires recovering candidate statements from their source context and checking whether they remain well posed and unresolved. We draw on search and recommender systems to organize this process. Building a pool of attemptable problems is akin to candidate retrieval, i.e., recovering statements from a large corpus using imperfect signals and filtering them for provenance, well-posedness, and current status (Belkin & Croft 1992; Liu 2009; Liu et al. 2022). We also draw on the idea of recommendation cascades, in which progressively more selective stages reduce a large collection of candidates to a small set of artifacts for expert attention (Ricci et al. 2010; Covington et al. 2016; Wang et al. 2011; Chen et al. 2017; Zhu et al. 2026). Our techniques also relate to literature-based discovery, which searches published knowledge for research opportunities not visible from a single paper (Swanson 1986b; Swanson 1986a). Our objective differs from automated conjecturing, a complementary line of work that creates new mathematical conjectures rather than surfacing existing ones. This line includes automated theory formation, the Ramanujan Machine, and the data-driven TxGraffiti system (Colton 2012; Raayoni et al. 2021; Davila 2026). Recent LLM work has used generated conjectures to expand formal training data and couple conjecturing with proving, as in LeanConjecturer and STP, while Moonshine makes conjecture generation the organizing objective of an autonomous mathematical research agent (Onda et al. 2025; Dong & Ma 2025; Chen & Jiang 2026). We instead recover unresolved statements that authors have already placed in the literature. Each candidate retains its source paper, statement text, local context, and status evidence, so that later attempts and reviews can be checked against what the source actually claimed. After status checking, the result is an attemptable pool P of source-grounded conjectures that appear well posed and still open. 2.3 Effort Allocation under Uncertainty Let U be the universe of mathematical questions that can be expressed in natural language. The human practice of mathematical research can be viewed as a large effort-allocation process over U. Mathematicians search this space for questions worth exploring, and then try to answer them or make progress on them. As AI agents become primary sources of mathematical attempts, allocating agent compute raises a similar problem. At the same time, the Leiden Declaration emphasizes that mathematicians remain responsible for validating and reporting mathematical claims (Alper et al. 2026), so the allocation of expert review effort also matters. The attemptable pool P described in Section 2.2 forms a small part of U, but it is a natural starting point because its conjectures and open problems have already been selected, stated, and discussed by mathematicians. Even for human mathematicians, deciding where to spend effort is difficult. Prior impressions of difficulty often differ from difficulty in hindsight. Some simply stated problems resist solution for a long time, while some long-standing problems eventually admit unexpectedly simple arguments. In an AI-assisted setting there is an additional source of uncertainty: what is difficult for human mathematicians need not be difficult in the same way for the current model. In turn, an important question is how to use limited model attempts to discover which conjectures from the attemptable pool P are likely to produce artifacts that are worth expert review. Figure 2: Schematic view of the current reachable region. The figure is illustrative, and both axes should be read qualitatively. A single pass over the pool probes which conjectures can produce artifacts worth review under the current model. As model capability improves, more conjectures may become reachable. The question of allocating limited attempts to a set of problems invites a simple bandit interpretation (Bubeck & Cesa-Bianchi 2012; Lattimore & Szepesvári 2019). From this perspective, a conjecture c∈c is an arm, and spending reasoning and review effort on it is a pull. A pull yields an assessed outcome Y(c)Y(c), which may be no reliable result, a known resolution, or a candidate proof or counterexample. In our experiments we pull each arm only once, as in the initialization stage of the UCB algorithm (Auer et al. 2002), and then use subsequent steps to concentrate human review on a much smaller set of outputs. Further investigating bandit algorithms for our setting is left for future work. Our effort allocation problem differs from the systems discussed in Section 2.1, which amount to concentrating reasoning effort on one or a few problems that are selected in advance. Finally, we note that effort allocation is dynamic: as models improve, conjectures that previously produced no useful progress may enter the current system’s reachable region, thereby demanding different allocations of effort than the older models. Figure 2 illustrates this reachable region. 3 Method Our method Find, Attempt, and Recommend (FAR) instantiates the allocation view above as a literature-to-review pipeline. Analogous to search and recommender systems (Wang et al. 2011; Chen et al. 2017; Liu et al. 2022, e.g.,), FAR progressively narrows a large collection through increasingly costly stages, with later stages receiving more compute per item. As illustrated in Figure 3, given a literature corpus and a research direction, FAR finds relevant open problems and filters them into an attemptable pool P (the Label, Extract, and Check stages), attempts candidate resolutions (Solve), and recommends selected conjecture-resolution pairs for expert review (Judge). Figure 3: From papers to recommendations for expert review. The upper row shows a search or recommender pipeline that recalls and filters candidates from a large corpus. Its numbers indicate typical orders of magnitude. The lower row shows the analogous FAR pipeline. Numbers in the lower row are counts from our pilot run detailed in Section 4. 3.1 Finding Relevant Open Problems The finding stage takes a literature corpus and a research direction as input and returns an attemptable pool P of relevant open problems. It identifies papers in the research direction, extracts unresolved statements from them, and finally checks whether those statements are valid and still open. The corpus may be arXiv, a topic-specific collection, or another large-scale source of mathematical literature. The prompts for the three stages are given in Appendix A. Finding relevant papers via labeling. Before the pipeline runs, a mathematician fixes a research direction at whatever granularity they need. An agent then reads each paper, labels whether it lies in that direction, and keeps the papers that do. Labeling plays the part of recall in a retrieval cascade. It sees the entire corpus, so it runs the cheapest model in the pipeline. Extracting and recovering conjectures. An agent extracts unresolved statements from the papers that survive labeling. A paper may yield several such statements or none. These may be labeled conjectures, questions, or open problems, or they may appear in prose. The extractor excludes future work that poses no specific mathematical question, and statements resolved within the same paper. Beyond those exclusions it is permissive, since a statement it passes over cannot be recovered later while a spurious one is dropped by the next stage. Each extracted statement becomes a candidate, recorded with its source paper. Table 1 gives a representative source-to-candidate example. Checking validity and status. An agent searches for later work on each candidate and records the supporting sources as status evidence. It labels the candidate open when its source paper states a concrete unresolved problem and no credible resolution is found, solved when credible evidence resolves it, and invalid when the extracted text does not state a concrete open problem. Open candidates form P, meaning that the pool contains conjectures that appear relevant, well posed, and still open under the available evidence. Like the eligibility filters that the drop items that a recommender can no longer serve, this stage removes the candidates that turn out to be solved or invalid. Table 1: A recovered candidate, from source text to the pool. Field Value Source paper Ikenmeyer, Pak, and Panova, Positivity of the Symmetric Group Characters is as Hard as the Polynomial Time Hierarchy Extracted label Conjecture 5.3.2 Extracted statement The problem ComputeCharBinary is GapP-complete under many-one reductions. Status Open. The check found no credible resolution, and records that the completeness question is still unsettled. 3.2 Attempting for Candidate Resolutions Every conjecture in the pool P receives one attempt, which is the unit of effort that this stage allocates. An attempt may be a single agent run, a longer harness, a multi-agent workflow, or repeated sampling under a selection rule. Performing one attempt per conjecture simply allocates the attempt budget uniformly across problems in the pool. As discussed in Section 2.3, this can be seen as the initialization step of a bandit algorithm, and we study other strategies in Section 4.3. To perform an attempt, an agent is given each conjecture as extracted, together with its source paper, so that its notation is read against the text that introduced it. Although the checking stage already searched for whether the problem is open, the checking stage ran a weaker model. Determining that a conjecture is equivalent to something that is already settled can take non-trivial reasoning, and hence may benefit from the stronger model used in the attempting stage. The agent therefore searches before it attempts, and labels the outcome KNOWN when a credible source already resolves the conjecture, NEW when it produces a complete proof or counterexample of its own, FIX when the statement as written is defective and the minimal repair it proposes cannot be settled, and NONE when it reaches none of these. The prompt is given in Appendix A. Only NEW outcomes go on to the next stage. The find stages narrowed down the pool of problems, meaning that the attempt stage can use more resources per problem. For the attempt stage we run the most capable model in the pipeline, and it returns a set Y pairing every conjecture in P with an outcome. 3.3 Recommendation for Expert Review Expert review is the scarcest resource in the pipeline, and only a few of the conjecture-resolution pairs in Y can receive it. The recommend stage decides which pairs receive expert review. It judges whether the result is correct and whether it is significant enough to publish, similar to what a referee determines in human peer review. The judging and grading in this stage are akin to the final filters in a search or recommender cascade. Here, a mathematician is the user that the results are served to. Judging. One or more agents check each NEW outcome in Y for correctness, asking whether it addresses the statement that it targets, whether the argument or construction is complete, and whether every step holds. Each agent marks the outcome PASS or FAIL, and an outcome passes only if every agent passes it. Recommending for review. A second agent takes the outcomes that passed and sorts each one as already known, as new but too minor to stand alone, or as substantial enough to publish on its own. Deciding the first requires a fresh literature search, since an existing resolution may have escaped both earlier stages. The last group forms the set A of artifacts. The judging and grading prompts are given in Appendix A. Expert review. By this point A is small, and everything in it lies in the direction that the mathematician set at the start. The mathematician(s) read the artifacts that interest them, check that each is correct and not already known, and write up those that hold. 4 A Pilot Run in Combinatorics We instantiate Find, Attempt, and Recommend in combinatorics, a domain whose results the authors have the expertise to verify. The remainder of this section reports the setup of the run, the outcomes it produced, and an analysis of those outcomes. 4.1 Setup For this pilot we assemble a corpus of 51,110 mathematics papers from OpenAlex metadata and source links (Priem et al. 2022). The pipeline is not tied to that source, and arXiv or another large-scale collection of mathematical literature would serve as well. Labeling keeps 5,245 papers. Extraction recovers 6,453 candidates from 2,742 of them, and checking leaves 4,717 conjectures drawn from 2,206 papers. These form the attemptable pool P. The stages run different models. Labeling uses gpt-oss-120b, extraction gemini-3.5-flash, and checking gemini-3.1-pro with web search. Attempting, judging, and grading all use gpt-5.5 at xhigh reasoning effort, and each conjecture in P received one attempt, instantiated here as a single run of the opencode agent in a working directory holding the paper and the statement. Each claimed resolution was put to three independent judges. This is the ordering Section 3 describes, with progressively more capable models as the set narrows and the task becomes harder. 4.2 Outcomes Every conjecture in P receives one attempt and one of four outcomes. Only NEW outcomes are judged, and only those that pass are graded. The tree below gives the count at each step. 4,717conjectures attempted==P2,905NONEno result443KNOWNalready resolved in the literature319FIXstatement defective as written1,050NEWclaimed resolution452FAILfails judging598PASSpasses judging75already known446too minor to stand alone77publishable==J==A The run produced claimed resolutions for 1,050 of the 4,717 conjectures. Judging accepted 598 of them, which we write J, and grading left 77 of those to form A. The authors reviewed 15 of the artifacts in A that they found particularly interesting, and every one of them is mathematically correct. One, an asymptotic bound on a divisor-difference problem of Erdős recorded in Guy’s miscellany, had been settled a few months before our run by a route that none of the searches in the cascade turned up. Section 5 discusses several of these results, and Appendix C collects the write-ups. 4.3 Analysis: Allocating Effort for Discovery Beyond the mathematics that the run produced, the outcomes of the run let us study alternative effort allocation strategies to the uniform strategy that we used in the pilot. To do so, we use the outcomes of each stage (namely, the conjectures J that ended up passing judging and the final artifacts A) along with difficulty and importance scores that were collected during the run. Concretely, the agent’s prompt in the checking stage asked it to include two estimates: • a difficulty d, anchored at 00 for a conjecture whose resolution would be an unpublishable exercise and at 11 for one publishable in a top journal; • an importance i, anchored at 00 for a statement with no substantive mathematical content and at 11 for a Fields-Medal-level problem. In both cases the prompt asks for the scores to follow a roughly normal distribution centered on 0.50.5. The scores are fixed before any reasoning budget is spent, and the run then attempted every conjecture in P, so the two scores can be read against outcomes that they preceded. 4.3.1 Validity of the Agent’s Difficulty and Importance Scores We first study the validity of the difficulty and importance scores, before using them within allocation strategies. For a group of conjectures ⊆S , write • δ()=|∖|/||δ(S)=|S |\,/\,|S|, the fraction of S with no accepted resolution; • ι()=|∩|/|∩| (S)=|A |\,/\,|J |, the fraction of its accepted resolutions graded publishable. The δ(S)δ(S) metric is viewed as an empirical estimate of the difficulty, and ι(S) (S) as an empirical estimate of the importance based on the outcomes of the run. Therefore, we use these metrics to validate the model-estimated difficulty score d and importance score i. The tree in Section 4.2 gives both over the whole pool, δ()=4,119/4,717δ(P)=4,119/4,717 and ι()=77/598 (P)=77/598. Reading d and i as maps →[0,1]P→[0,1], Figure 4 plots δ(d−1[a,b))δ(d^-1[a,b)) in (a) and ι(i−1[a,b)) (i^-1[a,b)) in (b), over the intervals [a,b)[a,b) marked on each axis. (a) difficulty d against δ (b) importance i against ι Figure 4: Each score against the quantity it judges. Panel (a) plots δ(d−1[a,b))δ(d^-1[a,b)) on an axis starting at 50%50\%, panel (b) plots ι(i−1[a,b)) (i^-1[a,b)). n counts attempts in (a) and accepted resolutions in (b). Candidates that a later status recheck reclassified as solved or invalid are excluded. Figure 4 shows that δ, the fraction of attempts with no accepted resolution, and ι , the publishable fraction of the accepted ones, both correlate positively with the difficulty and importance scores the agent assigned during the checking stage. We measure each association by the area under the ROC curve (AUC) (Hanley & McNeil 1982), the probability that the score ranks a randomly chosen conjecture that has the outcome above a randomly chosen conjecture that does not, ties counting half. Difficulty achieves an AUC of 0.690.69 against having no accepted resolution, and importance 0.600.60 against being graded publishable among the accepted. Appendix B.1 repeats the figure with intervals attached and gives these statistics in full. A Mann-Whitney test (Mann & Whitney 1947) puts the first at p<10−40p<10^-40 and the second at p=0.008p=0.008, both showing strong positive correlation. We therefore use them below in analyzing allocation strategies. We also note that the two scores are not independent of each other. Their Spearman rank correlation is 0.830.83, and the figure below illustrates that dependence. Over 60% of the conjectures lie strictly above the diagonal, while almost nothing falls strictly below. In other words, a problem that the agent calls important is almost always one that it also calls hard. This reflects a selection effect, since the pool holds only problems that are still open, and an important question stays open only while it remains hard. The two scores do not carry the same information, however. We demonstrate this by comparing each conjecture only against others that received the same importance score. For each value v of i, write AUCvAUC_v for the AUC of d within the conjectures scored v, and nvn_v for the number of pairs it compares. The stratified AUC is then, AUCstrat=∑vnvAUCv∑vnv=0.56,p<10−5.AUC_strat= _vn_v\,AUC_v _vn_v=0.56, p<10^-5. Therefore, among conjectures the LLM scored equally important, the difficulty score still effectively separates the attempts that returned an accepted resolution from those that did not. 4.3.2 Where to Spend a Limited Budget Section 2.3 posed the question: once agents are a primary source of mathematical attempts, how should their compute be allocated? We take it up here in a simple setting. Suppose the attempting stage of Section 3.2 is given a budget of B attempts, and each conjecture receives at most one attempt. Writing S for the set of conjectures that the stage attempts and f for the value of the artifacts that it produces, the stage should maximize the following: maximize⊆ _\;S [f(∩)] [\,f(A )\, ] (1) subject to to ||=B, |S|=B, where the expectation is over the outcomes of the attempts. Three choices of f are natural: • f1=|∩|f_1=|A |: the total number of publishable artifacts. • f2=∑c∈∩i(c)f_2= _c i(c): the total importance of those artifacts. • f3=maxc∈∩i(c)f_3= _c i(c): the importance of the single best artifact. Optimal strategies. Consider the outcome of a single attempt on conjecture c as random. Then δ(c)E\,δ(c) is the probability that it returns no accepted resolution, ι(c)E\, (c) the probability that a resolution it does return is graded publishable, and p(c)=(1−δ(c))ι(c)p(c)= (1-E\,δ(c) )\,E\, (c) the probability that c ends in A. If δ(c)E\,δ(c) and ι(c)E\, (c) are known in advance, we can solve the optimization problem (1) exactly for f1f_1 and f2f_2, and within a constant factor for f3f_3. For f1f_1 and f2f_2, by linearity of expectation: [f1(∩)]=∑c∈p(c),[f2(∩)]=∑c∈i(c)p(c).E\,[f_1(A )]= _c p(c), \,[f_2(A )]= _c i(c)\,p(c). Sorting P by p(c)p(c), respectively by i(c)p(c)i(c)\,p(c), and keeping the first B therefore solves (1). The argument for f3f_3 runs through the classical problem of monotone submodular maximization (Krause & Golovin 2014). Formally, a set function F on subsets of P is submodular if F()+F()≥F(∪)+F(∩)F(S)+F(T)≥ F(S )+F(S ) for all ,⊆S,T , and monotone if F()≤F()F(S)≤ F(T) whenever ⊆S . Maximizing a monotone submodular F under a cardinality constraint contains maximum coverage as a special case and is therefore NP-hard. It is known, however, that a simple greedy algorithm achieves at least a (1−1/e)(1-1/e) fraction of the optimum (Nemhauser et al. 1978), and that no polynomial algorithm beats it unless P=\,=\,NP (Feige 1998). The algorithm starts with 0=∅S_0= , and at each iteration j it adds the element of largest gain F(c∣j−1)=F(j−1∪c)−F(j−1)F(c _j-1)=F(S_j-1∪\c\)-F(S_j-1), so that j=j−1∪argmaxc∈∖j−1F(c∣j−1),j=1,…,B.S_j=S_j-1∪ \ *arg\,max_c _j-1F(c _j-1) \, j=1,…,B. We show that ↦[f3(∩)]S \,[f_3(A )] is monotone submodular, which is what the guarantee above requires. The map f3f_3 is monotone by definition. For submodularity, suppose without loss of generality that f3()≥f3()f_3(U)≥ f_3(V); then f3(∪)=f3()f_3(U )=f_3(U) and f3(∩)≤f3()f_3(U )≤ f_3(V), so f3()+f3()≥f3(∪)+f3(∩)f_3(U)+f_3(V)≥ f_3(U )+f_3(U ). The map ↦∩S preserves unions and intersections, so it carries both properties over to ↦f3(∩)S f_3(A ). Taking expectations then gives the same two properties for ↦[f3(∩)]S \,[f_3(A )]. Building a strategy from the scores. In practice, neither δ(c)E\,δ(c) nor ι(c)E\, (c) are known. Before any reasoning is spent, the only signals available are the model-based difficulty and importance scores that the pool building stage recorded. By the arguments above, if we can approximate p from these scores then we obtain near-optimal strategies for f1f_1 and f2f_2. For f3f_3 an approximation of p is not enough, since each greedy step needs the value of F(c∣)F(c ), which depends on the joint distribution of the attempt outcomes. Hence we propose an alternative algorithm for f3f_3 that, like those for f1f_1 and f2f_2, needs only an approximation of p and does well in our run. We fit the two factors of p by least squares, 1−δ1-δ on d over P and ι on i over J, and write δ δ and ι for the resulting estimates of δE\,δ and ιE\, . We write δ δ and ι for the two estimates of δE\,δ and ιE\, . Their product p^=(1−δ^)ι p=(1- δ)\, estimates p. In theory, ranking P on p p and taking the top B is near-optimal for f1f_1, and ranking on ip^i\, p is near-optimal for f2f_2. For f3f_3, we choose a top fraction of P based on importance, and then rank on p p. The least squares fits underlying these strategies do not require the whole pool, so a round may attempt a small part of the pool first, fit and compare strategies on the returned outcomes, and allocate the remaining budget with the strategy that performed best. Empirical results. We compare how these strategies perform on the outcomes that our pilot run produced. As the baseline strategy, we take S to be a uniform random subset of P of size B. Against it we compare ranking the whole of P on p p, ranking it on ip^i\, p, and ranking on p p inside a top fraction of P by importance (inside such a restriction i varies too little to reorder anything, so ranking there on ip^i\, p gives an almost same strategy). Figure 5 shows the baseline, the two rankings, and the two restrictions that returned the most artifacts. Each strategy is evaluated by 55-fold cross-validation. We split P into five parts at random, fit both factors on four of them, rank the fifth by the resulting p p and keep its top B/5B/5, and combine the five selections, so that no conjecture is ever ranked by a fit that saw its own outcome. Each point of the figure averages 10001000 such partitions, at B=10B=10, 2525, 5050, 7575, 100100, 200200 and 300300. Appendix B.2 tabulates every plotted point. Figure 5: Allocation strategies against the budget. Each fit is made on four fifths of P and applied to the remaining fifth, from which B/5B/5 conjectures are drawn. The five selections together make one set of B conjectures, and each point averages what that set returns over 10001000 random partitions. Ties are broken at random, and the uniform baseline is computed exactly from its closed form. Here, B only counts the allocated attempts. On f1f_1, the run follows the theoretical analysis closely. Ranking P on p p outperforms every other strategy at every budget. On f2f_2, the analysis prescribes ranking on ip^i\, p, but ranking on p p instead beats it narrowly at every budget. On f3f_3, neither ranking of the whole pool does well, and at the larger budgets both even fall behind the uniform random baseline. What works best in practice is to discard all but the most important part of P and rank what remains on p p. Of the fractions we tried, keeping the most important 1/101/10 did best on this run. Note that the results above depend on the scoring model, on the sources that the pool was built from, and on the models in the cascade. Nevertheless, the results suggest that using strategies derived from model-based importance and difficulty scores, it is possible to allocate the budget more effectively than with a uniform strategy in our problem setting. Furthermore, the optimal allocation strategy differs between f1f_1 and f3f_3, providing evidence that the best strategy can depend on the objective that the mathematician is optimizing for. 5 Selected Reviewed Results We manually reviewed fifteen of the artifacts from the pilot run, chosen by our own interest, and found no mathematical error in any of them. We discuss some of them below, which show three different degrees of novelty. Full write-ups of these results and of the others we reviewed are collected in Appendix C. 5.1 A Known Result Graded as New Sets with no large divisor difference (Appendix C.4). Erdős (Guy 1983) asks how large a set of integers can be if no two of its elements have a large difference dividing the larger one. Formally, for t≥1t≥ 1, let F(n,t)F(n;t) be the largest size of a set A⊆1,…,nA \1,…,n\ in which no two elements x<yx<y satisfy both (y−x)|y(y-x) y and y−x≥ty-x≥ t. The question is whether F(n,t)≤(12+o(1))nF(n;t)≤( 12+o(1))n for every fixed t. The artifact returned for this problem proves that F(n,t)/n→12F(n;t)/n→ 12. The odd numbers give the lower bound, since the difference of two odd numbers is even and cannot divide the larger. For the upper bound it fixes a finite set P of odd primes exceeding t with ∑p∈P1/p _p∈ P1/p large. For each p∈Pp∈ P the set cannot hold both 2rp2rp and (2r−1)p(2r-1)p, whose difference is p≥tp≥ t and divides 2rp2rp, so the even elements of A are divisible by no more primes of P, in total, than the odd integers outside A are. A second-moment estimate on each parity class turns this into the matching upper bound. The recommend stage judged the proof correct and graded it as a publishable result. Reviewing it ourselves we found the mathematics correct but the result already known. Four months before our run a proof of the same statement had been recorded on the Erdős problems site, obtained by Liam Price with ChatGPT-5.2 (Bloom 2026), and Tao also observed there that the bound follows quickly from an inequality of Elliott 2012. The cascade never found this record. 5.2 A Connection Not Previously Made Small unions of lines closing a route to the Nikodym bound (Appendix C.10). For a set L of affine lines in q3F_q^3 let P(L)=⋃ℓ∈LℓP(L)= _ ∈ L be the union of its lines. Lund et al. 2018 conjecture that for every constant C>0C>0 and every α with α(q)/q→∞α(q)/q→∞, a set L of at least Cq3Cq^3 lines in which no plane contains α(q)α(q) lines must satisfy |P(L)|≥(1−o(1))q3|P(L)|≥(1-o(1))q^3. They show that this would give an optimal bound on the size of a Nikodym set in three dimensions, and Tao 2025 still cites it as open in that role. The artifact returned for this problem disproves the conjecture for every odd q. It takes the affine paraboloid z=x2−νy2z=x^2-ν y^2 with ν a nonsquare, and keeps at each of its points the (q+1)/2(q+1)/2 tangent lines on which x2−νy2−zx^2-ν y^2-z is always a square. The family has q2(q+1)/2q^2(q+1)/2 lines, no affine plane holds more than q+1q+1 of them, and its union has exactly q2(q+1)/2q^2(q+1)/2 points, a density of 1/2+o(1)1/2+o(1). The same family also refutes the stronger Conjecture 1.5 of that paper. We found the construction mathematically correct, and verified the counts exhaustively for q≤13q≤ 13. The recommend stage, however, informed us that the construction already exists in other contexts. In projective language the family is a classical object of finite geometry, the half-tangent partition of an elliptic quadric (Bruen & Drudge 1999; Cossidente & Pavese 2017). The contribution here is to link the existing construction to this conjecture. 5.3 Results with No Precedent Found The three results below have each been checked by an author or by a domain expert, and are new so far as we could determine. They represent three kinds of discovery. The first proves a conjecture, the second refutes one by counterexample, and the third answers an open-ended question. Many-one GapP-completeness for binary symmetric group characters (Appendix C.6). For partitions λ and μ of an integer n, write χλ(μ)χ^λ(μ) for the irreducible character value of the symmetric group. Ikenmeyer et al. 2024 consider the problem of computing χλ(μ)χ^λ(μ) from λ and μ given as lists of parts in binary. They prove that it is GapP-complete under Turing reductions and conjecture that it is GapP-complete under many-one reductions. The artifact returned for it proves that conjecture. It reduces the difference of two counts of exact covers of a finite set to a single character value at a two-row partition λ=(n−s,s)λ=(n-s,s), and checks membership in GapP separately. A two-row character value is itself a difference, Nμ(s)−Nμ(s−1)N_μ(s)-N_μ(s-1), where Nμ(t)N_μ(t) is the number of ways to choose parts of μ summing to t. Maximum versus average independent set size in triangle-free graphs (Appendix C.3). For a graph G let α(G)α(G) be its independence number and α¯(G) α(G) the expected size of a uniformly random independent set. Davies et al. 2018 conjecture that α(G)/α¯(G)≥2−od(1)α(G)/ α(G)≥ 2-o_d(1) for every triangle-free G of minimum degree d, and show that this would give R(3,k)≤(12+o(1))k2/logkR(3,k)≤( 12+o(1))k^2/ k. It is restated as open in recent surveys of the hard-core model and Ramsey theory (Davies & Kang 2025; Morris 2026). The artifact returned as a counterexample the Cartesian product C5□Km,mC_5\, \,K_m,m, which is triangle-free and (m+2)(m+2)-regular. An independent set of the product picks an independent subset of C5C_5 at each vertex of Km,mK_m,m, and the two sides must pick disjoint subsets. This makes α=4mα=4m and the expected size computable exactly, and the ratio tends to 24/1324/13. Replacing C5C_5 by the circulant C13(1,5)C_13(1,5) further lowers the limit to 32/1932/19. Divisibility among binomial coefficients (Appendix C.5). For a fixed integer n≥2n≥ 2, Erdős & Straus 1977 ask for the natural density d∗(n)d^*(n) of those m that admit some k with 1≤k≤m−n1≤ k≤ m-n and (n+kn)|(m+k) n+kn m+kk, that is, the proportion of such m among the integers up to x as x→∞x→∞. They settle n=1n=1 themselves and record that n=2n=2 “seems much more difficult to decide”. The returned artifact shows that d∗(n)=1d^*(n)=1 for every fixed n≥2n≥ 2. Given a bound B>nB>n it takes kB=∏p≤Bpepk_B= _p≤ Bp^e_p with epe_p least such that pep>np^e_p>n, so that by Kummer’s theorem every prime factor of NB=(n+kBn)N_B= n+k_Bn exceeds B. Each such prime q therefore exceeds n, and so divides exactly one of kB+1,…,kB+nk_B+1,…,k_B+n, say kB+i(q)k_B+i(q). By Legendre’s formula the condition mmodq≥i(q)m q≥ i(q) makes (m+kBkB) m+k_Bk_B divisible by the full power of q that divides NBN_B. The Chinese remainder theorem turns these residue conditions into a set of density ∏q(1−i(q)/q) _q(1-i(q)/q) that tends to 11 as B→∞B→∞. 6 Conclusion We have argued that an important–yet understudied–aspect of AI for mathematics concerns selecting which problems to solve. We introduced FAR, a literature-to-review cascade that recovers open problems from a corpus, attempts each of them, and recommends promising research outcomes for expert review. In a combinatorics pilot, FAR recovered 6,453 candidate statements, checked 4,717 of them into an attemptable pool, and returned 77 artifacts graded as substantial enough to publish, 15 of which the authors reviewed and found to be correct. Using the outcomes of our run, we studied strategies for allocating effort, i.e., selecting a subset of conjectures to attempt given a budget. We framed this problem as constrained optimization and derived strategies that are favorable both theoretically and in practice. We showed that simply asking an agent in our pipeline to output difficulty and importance scores before any reasoning is spent yields scores that can be used effectively within the allocation strategies. We hope that FAR is a starting point for new techniques that help explore mathematics, which consists of not just solving problems, but of deciding which problems are worth spending effort on in the first place. Acknowledgments We thank Sylvester W. 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A.1 Finding Relevant Open Problems In the pilot the direction was “combinatorics”. ⬇ Return one JSON object with this schema: ”comment”: ”…”, ”in_direction”: false Rules: - ‘comment‘ must name the paper’s primary subject in a few words. - ‘in_direction‘ must be a JSON boolean. - Use ‘true‘ when the paper’s primary content lies in the research direction. - Use ‘false‘ when it does not, when the content is not mathematical, or when the paper appears mislabeled. Research direction: direction Paper content: text ⬇ Return one JSON object with this schema: ”title”: ”…”, ”authors”: [”…”, ”…”], ”decision_basis”: ”…”, ”has_open_conjecture”: false, ”conjectures”: [ ”conjecture_label”: ”…”, ”conjecture_text”: ”…”, ”conjecture_section”: ”…” ] Rules: - ‘title‘ must be a non-empty string. - ‘authors‘ must be a JSON array of non-empty author-name strings. - ‘decision_basis‘ must be one short English sentence. - ‘has_open_conjecture‘ must be a JSON boolean. - ‘conjectures‘ must be a JSON array. If ‘has_open_conjecture‘ is false, it must be ‘[]‘. - Set ‘has_open_conjecture‘ to true iff the paper contains at least one explicit unresolved mathematical statement. - Count these as hits: 1. labeled ‘Conjecture‘ / ‘Question‘ / ‘Open Problem‘ 2. sentences with markers like ‘open question‘, ‘open problem‘, ‘open issue‘, ‘remains unknown whether‘, or ‘we suspect … although we have been unable to establish …‘ 3. a direct statement that a specific mathematical property, existence claim, or classification problem ‘still remains an open issue‘ - Do NOT count: 1. generic future work that does not pose a specific mathematical question 2. results that have already been proved or resolved within the paper itself - If a sentence says a specific claim or property is ‘still an open issue‘, count it even if it is not written as a formal question. - If ‘has_open_conjecture‘ is true, extract only the explicit unresolved statements themselves, not nearby speculation. - ‘conjecture_label‘ should use the paper’s label when present, otherwise use a short fallback like ‘Unlabeled open problem 1‘. - ‘conjecture_text‘ should copy the paper’s unresolved statement as faithfully as possible and preserve notation. - ‘conjecture_section‘ should be the visible section/subsection title, or ‘”‘ if unavailable. Paper content: text ⬇ Return one JSON object with this schema: ”sources”: [ ”title”: ”…”, ”url”: ”…”, ”claim”: ”…” ], ”reason”: ”…”, ”status”: ”solved”, ”importance”: 0.5, ”difficulty”: 0.5 Rules: - Verify the candidate’s current status using current web information. - ‘status‘ must be one of: ‘open‘, ‘solved‘, ‘invalid‘. - Use ‘open‘ when the candidate is a concrete open problem in the source and no credible solved evidence is found. - Use ‘solved‘ when a credible source appears to solve it. - Use ‘invalid‘ when it is not a concrete open problem in the source. - ‘sources‘ should list only sources directly supporting the status. - each ‘claim‘ must be what that source says about this candidate. - for ‘solved‘, ‘sources‘ must name at least one source that resolves the candidate. - ‘reason‘ must be one concise English sentence. - ‘importance‘ must be a number in [0, 1] for the candidate itself: candidates with no substantive mathematical content should be scored 0; Fields-Medal-level problems should be scored 1; most ordinary research problems should follow a roughly normal distribution centered around 0.5. - ‘difficulty‘ must be a number in [0, 1]: solving it would be an unpublishable exercise should be scored 0; solving it would be publishable in a top journal (Annals, Inventiones, JAMS, Acta) should be scored 1; most problems should follow a roughly normal distribution centered around 0.5. - For ‘solved‘ or ‘invalid‘, set ‘importance‘ and ‘difficulty‘ to 0. Paper title: title Paper authors: authors Candidate label: conjecture_label Candidate section: conjecture_section Candidate text: conjecture_text A.2 Attempting for Candidate Resolutions ⬇ You are a research-level mathematical reasoner. This is a test to see how well you can craft non-trivial, novel and creative proofs given a math problem. Given a natural-language problem, conjecture, or paper metadata, reconstruct the most likely formal mathematical statement and resolve it. First, state the reconstructed conjecture precisely, including all hypotheses, definitions, notation, quantifiers, ambient category, and axiom system when relevant. Explain briefly what information supports this reconstruction. If the reconstruction is ambiguous, list the plausible formalizations and choose one to analyze, explicitly noting the ambiguity. Do not treat the fact that the source labels the statement open, conjectural, unresolved, or a problem as a reason to stop. The task is to attack the statement mathematically. However, do not lower the standard of proof. Never present an incomplete, heuristic, or speculative argument as a complete proof. Before committing to a proof, test the statement against degenerate, extremal, low-dimensional, finite, infinite, and standard model examples appropriate to the field. Look actively for counterexamples as well as proofs. If the literal statement is false because of a degenerate, boundary, vacuous, or typo-like case, do not stop after giving the counterexample. Instead: - State the literal counterexample clearly and explain why it falsifies the literal statement. - Diagnose whether the failure appears to come from a small formulation defect, such as a missing nonzero/nonempty/nontrivial assumption, a wrong inequality direction, an omitted endpoint condition, a missing connectedness or finiteness hypothesis, a confusion between strict and non-strict inequalities, a missing regularity condition, or a convention mismatch. - Propose the minimal natural repair or repairs to the statement, using the fewest and most standard changes consistent with the paper’s terminology, surrounding context, and apparent mathematical intent. - Check that the proposed repair is not merely ad hoc, vacuous, or so weakened that it no longer captures the intended conjecture. - Retest the repaired statement against the original counterexample and nearby degenerate cases. - Then prove or refute the most plausible repaired statement. A complete answer must be a rigorous proof or a rigorous counterexample. Present the reasoning in a locally checkable form: definitions, lemmas, propositions, and proofs. For every invoked theorem, verify its hypotheses in the present setting. Track dependencies of constants, choices, witnesses, bases, subsequences, exceptional sets, embeddings, isomorphisms, and parameters. If the proof or counterexample is known in the literature, state that honestly and provide a reliable reference. Distinguish exact resolutions from stronger theorems, weaker partial results, equivalent reformulations, and merely related work. Do not invent references. After the proof or counterexample, include a verification audit confirming that the formalized statement matches the reconstructed conjecture, that no extra assumptions were introduced, that all theorem hypotheses were checked, and that the conclusion exactly matches the target statement. Response format: The first line must be exactly one of: KNOWN, NEW, FIX, NONE. - KNOWN: a reliable existing source in the literature already proves the conjecture or gives a counterexample/disproof. Cite the source. - NEW: your answer gives a complete resolution that is not presented as known literature. Use NEW for either a complete proof that the conjecture is true or a complete counterexample/disproof that the conjecture is false. - FIX: you have identified a small formulation defect and proposed a minimal natural repair, but you are unable to prove or refute the repaired statement. Use FIX to indicate that you have done this. - NONE: you found neither a known resolution nor a reliable complete proof/counterexample despite all efforts. Then use these sections exactly: Problem: Result: Citation: ⬇ Read input.json in the current directory. It contains the paper title, authors, paper text, the sources a status check turned up, and target conjecture. The target conjecture is in conjecture.text. Resolve that target conjecture and return only the required labeled answer. A.3 Judging and Grading KNOWN and NEW outcomes are sent to the judge, and its label routes the outcome rather than certifying it. Only the results it accepts as new reach the grader. ⬇ You are a strict referee for natural-language mathematics proofs. This is a test to see how well you can referee a proposed natural-language mathematics proof given a math problem. Check the claimed resolution or disproof against the target conjecture supplied in the user task. Accept only if the claimed resolution or disproof attacks the correct statement and is mathematically rigorous and complete. A valid counterexample or disproof may pass if it rigorously disproves the conjecture. Reject if it has fatal proof gaps, hallucinated dependencies, hidden assumptions, or a mismatch between the stated theorem and the original conjecture. Do not reject merely because the original paper called the conjecture open. In the case when the claimed resolution or disproof is NEW, you should also conduct a very thorough literature search using the web search tool to see if a similar or stronger result already exists in the literature. On the first line, write exactly one word: PASS or FAIL or KNOWN. - PASS: the claimed resolution is mathematically complete and attacks the correct statement, and in the case of NEW, a similar or stronger result does not exist in the literature despite your best search efforts. - KNOWN: the claimed resolution is NEW, but a similar or stronger result already exists in the literature. - FAIL: if neither of the above conditions are met. Then briefly explain your verdict, including the most important gap if you fail it. ⬇ Read input.json and solution.md in the current directory. input.json contains paper metadata, the paper text, the sources a status check turned up, and the target conjecture. solution.md contains the claimed resolution to check. Return only PASS or FAIL or KNOWN followed by your explanation. ⬇ You are a senior combinatorics referee performing a final quality-control pass on a result that a prover produced and a judge already accepted as a correct resolution. Your job is NOT to re-verify correctness from scratch (assume the proof is correct unless a literature search clearly contradicts it). Your job is to classify the result by its novelty and publishable significance, so a human can triage it afterwards. Do two things: 1. Literature check. Conduct a very thorough web search to determine whether the resolution, or a similar or stronger statement, is already known in the literature. Go beyond just searching for papers that cite the original paper; you should search for all open-access notes, surveys, forums, and other sources that might contain the result. The prover and earlier judges may have missed an existing reference; catching such cases is a primary goal of this pass. 2. Significance grading. If the result is genuinely not in the literature, assess how significant it is as a contribution to combinatorics: how hard, how novel, how interesting to the community, and what venue it would plausibly merit. On the first line, write exactly one token: KNOWN, TYPE1, TYPE2, or TYPE3. - KNOWN: the result (or a similar or stronger result) is in fact already known in the literature, despite the prover and earlier judges treating it as new. Cite the reference. - TYPE1: genuinely new but minor and unpublishable on its own (e.g. a routine exercise, a trivial special case, an immediate corollary of standard results). - TYPE2: genuinely new and substantial enough to support a standalone paper in a standard combinatorics or mathematics journal. - TYPE3: genuinely new and strong enough to merit publication in a top combinatorics journal (a major advance, a resolved well-known conjecture, or a result of broad interest). These boundaries are deliberately rough; when uncertain between two grades, pick the lower one and explain the uncertainty. After the first line, use these sections exactly: Classification rationale: Literature check: Citation: ⬇ Read input.json, solution.md, and judge.md in the current directory. input.json contains the paper metadata, the paper text, the sources a status check turned up, and the target conjecture, solution.md contains the resolution that was accepted as new, and judge.md contains the verdicts of the earlier judges. Classify the result and return only KNOWN, TYPE1, TYPE2, or TYPE3 on the first line, followed by the required sections. The artifacts put forward for expert review are the TYPE2 and TYPE3 items. Appendix B Analysis Details This appendix gives the data behind the two figures of Section 4.3. B.1 Score validity Figure 6 gives the same two associations as Figure 4 at a uniform bin width of 0.10.1. Each point is the measured rate in its bin and each bar its 95%95\% Wilson interval. Figure 6: Each score against the quantity it judges, with Wilson intervals. Panel (a) gives δ against the difficulty score, panel (b) gives ι against the importance score. B.2 Allocation curves The tables below give the value of each point plotted in Figure 5. Every entry is an average over 10001000 random partitions of the pool into five parts, with both factors of p p fitted on four of them and the remaining part ranked by the result, from which B/5B/5 conjectures are taken. The five selections together form the set of B conjectures that the entry scores. Ties are broken at random, and the uniform baseline is evaluated from its closed form rather than sampled. strategy B=10B=10 B=25B=25 B=50B=50 B=75B=75 B=100B=100 B=200B=200 B=300B=300 uniform random 0.17 0.43 0.86 1.29 1.72 3.44 5.17 rank on p p 0.40 0.97 1.64 2.29 2.95 5.42 7.92 rank on ip^i\, p 0.15 0.47 1.02 1.59 2.24 4.78 7.25 rank on p p inside the top 1/101/10 0.30 0.70 0.99 1.01 1.06 1.99 2.00 rank on p p inside the top 1/51/5 0.17 0.42 0.86 1.25 1.68 2.99 3.72 Table 2: Expected number of artifacts, the objective f1f_1. strategy B=10B=10 B=25B=25 B=50B=50 B=75B=75 B=100B=100 B=200B=200 B=300B=300 uniform random 0.08 0.21 0.42 0.63 0.84 1.68 2.51 rank on p p 0.16 0.40 0.73 1.05 1.38 2.62 3.87 rank on ip^i\, p 0.09 0.28 0.60 0.89 1.22 2.49 3.72 rank on p p inside the top 1/101/10 0.26 0.60 0.84 0.86 0.90 1.69 1.70 rank on p p inside the top 1/51/5 0.13 0.34 0.69 1.00 1.34 2.44 3.06 Table 3: Expected total importance of the artifacts returned, the objective f2f_2. strategy B=10B=10 B=25B=25 B=50B=50 B=75B=75 B=100B=100 B=200B=200 B=300B=300 uniform random 0.078 0.176 0.299 0.386 0.448 0.574 0.629 rank on p p 0.157 0.354 0.437 0.467 0.485 0.499 0.500 rank on ip^i\, p 0.090 0.280 0.546 0.597 0.600 0.600 0.600 rank on p p inside the top 1/101/10 0.259 0.596 0.841 0.850 0.850 0.850 0.850 rank on p p inside the top 1/51/5 0.126 0.294 0.540 0.690 0.779 0.850 0.850 Table 4: Expected maximum importance among the artifacts returned, the objective f3f_3. Appendix C Reviewed Solutions This appendix collects the write-ups of the results the authors reviewed, ordered alphabetically by the mathematicians who posed the problems. C.1 Paley graphs with a prescribed adjacency property This problem concerns the number of vertices of a Paley graph that are adjacent to one prescribed vertex and to neither of two others. Ananchuen and Caccetta proved that the Paley graph on q vertices has at least k such vertices, for every choice of the three, as soon as q>(1+22k)2q>(1+2 2k)^2, and conjectured that this threshold is exact. We show that it is not: the Paley graph on 55=31255^5=3125 vertices has at least 377377 such vertices for every choice, although 3125<(1+2754)23125<(1+2 754)^2. The proof is a quadratic character count in which the only obstruction is an elliptic curve over 55F_5^5 of trace 110110, and Waterhouse’s classification of elliptic-curve traces forbids a nonzero trace divisible by the characteristic over a field of odd degree in characteristic greater than 33. C.1.1 Introduction Let m,nm,n be nonnegative integers and let k be a positive integer. Following Ananchuen et al. 1992, a graph G has property P(m,n,k)P(m,n,k) if for every pair of disjoint sets A,B⊆V(G)A,B V(G) with |A|=m|A|=m and |B|=n|B|=n there are at least k vertices outside A∪BA∪ B that are adjacent to every vertex of A and to no vertex of B. We write (m,n,k)G(m,n,k) for the class of graphs with property P(m,n,k)P(m,n,k). These classes are quantitative refinements of the adjacency axioms of Blass & Harary 1979, who showed by a probabilistic argument that almost all graphs have property P(n,n,1)P(n,n,1), from which the same follows for every P(m,n,k)P(m,n,k); a graph is n-existentially closed exactly when it lies in (i,n−i,1)G(i,n-i,1) for all 0≤i≤n0≤ i≤ n. Explicit examples are much harder to come by, and the extremal question of how few vertices a graph in (m,n,k)G(m,n,k) can have was raised by Exoo 1981 and studied systematically by Ananchuen et al. 1992. For a prime power q≡1(mod4)q≡ 1 4, the Paley graph GqG_q has vertex set qF_q, two distinct vertices u,vu,v being adjacent exactly when u−vu-v is a nonzero square in qF_q; this is well defined because −1-1 is a square in qF_q. Paley graphs are the standard supply of explicit graphs with prescribed adjacency properties. Blass et al. 1981 showed that Gp∈(n,n,1)G_p (n,n,1) for every prime p≡1(mod4)p≡ 1 4 with p>n224np>n^22^4n, and Ananchuen & Caccetta 1993 obtained thresholds for the full range of parameters by estimating the relevant character sums, among them q>(1+22k)2⟹Gq∈(1,2,k).q> (1+2 2k )^2 G_q (1,2,k). This same threshold reappears in Ananchuen & Caccetta 1995, where it is deduced, together with the companion conclusion Gq∈(2,1,k)G_q (2,1,k), from the n-parameter implication q>(n−2)2n+2q+(n+2k−1)2n−2n−1⟹Gq∈(1,n,k)∩(n,1,k),q> \(n-2)2^n+2 \ q+(n+2k-1)2^n-2n-1 G_q (1,n,k) (n,1,k), which at n=2n=2 reduces to the threshold above. For n=1n=1 the n-parameter threshold is exactly right. Indeed it then reads q>4k−3q>4k-3, which for q≡1(mod4)q≡ 1 4 means q≥4k+1q≥ 4k+1, while Exoo 1981 proved that every graph in (1,1,k)G(1,1,k) has at least 4k+14k+1 vertices; hence Gq∈(1,1,k)G_q (1,1,k) if and only if q≥4k+1q≥ 4k+1 (Ananchuen & Caccetta 1995). Ananchuen & Caccetta 1995 leave the analogous exactness for n=2n=2 as a conjecture, writing: We have verified, by computer, that if q≡1(mod4)q≡ 1 4 is a prime power less than or equal to 10091009 and k is a positive integer with q<(1+22k)2q< (1+2 2k )^2, then Gq∉(1,2,k)G_q (1,2,k). We conjecture that this is true for all q. It is convenient to restate the conjecture as a single inequality. The Paley graph is vertex-transitive under translation, so in testing property P(1,2,k)P(1,2,k) we may always take A=0A=\0\. Writing η for the quadratic character of qF_q, extended by η(0)=0η(0)=0, the number of vertices to be counted for B=b,cB=\b,c\ is N(b,c)=#x∈q∖0,b,c:η(x)=1,η(x−b)=η(x−c)=−1.N(b,c)=\# \x _q \0,b,c\:η(x)=1,\ η(x-b)=η(x-c)=-1 \. (2) Thus Gq∈(1,2,k)G_q (1,2,k) if and only if k≤Nmin(q)k≤ N_ (q), where Nmin(q)=minN(b,c):b,c∈q∗,b≠c.N_ (q)= \N(b,c):b,c _q^*,\ b≠ c \. Since q<(1+22k)2q<(1+2 2k)^2 is equivalent to 8k>(q−1)28k>( q-1)^2, the conjecture asserts precisely that 8Nmin(q)≤(q−1)2for every prime power q≡1(mod4).8\,N_ (q)≤( q-1)^2 every prime power q≡ 1 4. (3) Ananchuen and Caccetta go on, in the same remark, to choose the three vertices in their character-sum estimate so that it yields an upper bound as well, concluding that Gq∉(1,2,k)G_q (1,2,k) whenever q<(−1+22(k+1))2q< (-1+2 2(k+1) )^2. The conjecture, they note, therefore has content only in the window (−1+22(k+1))2≤q≤(1+22k)2, (-1+2 2(k+1) )^2≤ q≤ (1+2 2k )^2, an interval of length (8+o(1))2k(8+o(1)) 2k. Our counterexample lies inside it: for k=377k=377 the window is 2915.01…≤q≤3126.83…2915.01…≤ q≤ 3126.83…, and q=3125q=3125. Ananchuen 2001 and Ananchuen & Caccetta 2006 carry the character-sum method of the note (Ananchuen & Caccetta 1995) over to the graphs built from cubic and quartic residues, and Bonato 2009 surveys the explicit constructions of n-existentially closed graphs. The note’s companion paper (Australia 1994) takes up (1,2,1)G(1,2,1) directly, exhibiting a graph in that class of every order at least 1010 except 1111 and noting that (1,2,1)G(1,2,1) contains no graph of any other order. We are aware of no earlier counterexample to the conjecture and, beyond the two bounds above, of no partial result on the exactness of the threshold for (1,2,k)G(1,2,k). Theorem C.1.1. The Paley graph G3125G_3125 on 555^5 vertices belongs to (1,2,377)G(1,2,377), while 3125<(1+22⋅377)2.3125< (1+2 2· 377 )^2. In particular equation 3 fails at q=55q=5^5, so the conjecture of Ananchuen & Caccetta 1995 is false. Remark C.1.2. The constant 377377 is best possible for this q. Since x5−x+1x^5-x+1 is irreducible over 5F_5, we may realize 3125F_3125 as 5[a]F_5[a] with a5=a−1a^5=a-1; take b=1b=1 and c=4+4a+a3c=4+4a+a^3. In the notation of Lemma C.1.3 below, b and c are squares while b−cb-c is not, so R(b,c)=8R(b,c)=8, and a computation in 55F_5^5 gives S(b,c)=−102S(b,c)=-102; hence 8N(b,c)=3126−102−8=3016.8N(b,c)=3126-102-8=3016. Therefore N(b,c)=377N(b,c)=377, which with Theorem C.1.1 gives Nmin(3125)=377N_ (3125)=377 and G3125∉(1,2,378)G_3125 (1,2,378). The margin is thin: 8Nmin(3125)=30168\,N_ (3125)=3016 exceeds (3125−1)2=3014.19…( 3125-1)^2=3014.19… by less than 22. Expanding the three character conditions in equation 2 turns 8N(b,c)8N(b,c) into q+1+S(b,c)−R(b,c)q+1+S(b,c)-R(b,c), where S(b,c)=∑x∈qη(x(x−b)(x−c))S(b,c)= _x _qη (x(x-b)(x-c) ) is minus the trace of the elliptic curve y2=x(x−b)(x−c)y^2=x(x-b)(x-c) and R(b,c)∈0,4,8R(b,c)∈\0,4,8\ is a correction coming from the three points 0,b,c0,b,c. For q=3125q=3125 Hasse’s bound gives S(b,c)≥−111S(b,c)≥-111 and hence N(b,c)≥376N(b,c)≥ 376, one short of what is needed. The crucial observation is that the equality case N(b,c)=376N(b,c)=376 forces the curve to have trace exactly 110110, a nonzero multiple of the characteristic, and over 55F_5^5 no elliptic curve has such a trace. C.1.2 The character count Throughout the rest of this section q is a prime power with q≡1(mod4)q≡ 1 4, and η is the quadratic character of qF_q, extended by η(0)=0η(0)=0. Thus η(x)=1η(x)=1 if x is a nonzero square, η(x)=−1η(x)=-1 if x is a nonsquare, and η(−1)=1η(-1)=1. In particular distinct u,v∈qu,v _q are adjacent in GqG_q if and only if η(u−v)=1η(u-v)=1. Only one character sum evaluation is needed: ∑x∈qη((x−u)(x−v))=−1(u≠v). _x _qη ((x-u)(x-v) )=-1 (u≠ v). (4) Indeed, substituting x=u+(v−u)yx=u+(v-u)y turns the left side into ∑yη((v−u)2y(y−1))=∑yη(y(y−1)) _yη ((v-u)^2y(y-1) )= _yη (y(y-1) ). The term y=0y=0 vanishes, and for y≠0y≠ 0 we have η(y(y−1))=η(y2(1−y−1))=η(1−y−1)η (y(y-1) )=η (y^2(1-y^-1) )=η(1-y^-1). As y runs over q∗F_q^* the element 1−y−11-y^-1 runs over q∖1F_q \1\, so the sum equals ∑w≠1η(w)=−η(1)=−1 _w≠ 1η(w)=-η(1)=-1. Lemma C.1.3. Let b,c∈q∗b,c _q^* be distinct and let N(b,c)N(b,c) be as in equation 2. Then 8N(b,c)=q+1+S(b,c)−R(b,c),8N(b,c)=q+1+S(b,c)-R(b,c), (5) where S(b,c)=∑x∈qη(x(x−b)(x−c))S(b,c)= _x _qη (x(x-b)(x-c) ) and R(b,c)=(1−η(b))(1−η(c))+(1+η(b))(1−η(b−c))+(1+η(c))(1−η(b−c)).R(b,c)= (1-η(b) ) (1-η(c) )+ (1+η(b) ) (1-η(b-c) )+ (1+η(c) ) (1-η(b-c) ). Moreover R(b,c)∈0,4,8R(b,c)∈\0,4,8\. Proof. Put T=∑x∈q(1+η(x))(1−η(x−b))(1−η(x−c)).T= _x _q (1+η(x) ) (1-η(x-b) ) (1-η(x-c) ). For x∉0,b,cx∉\0,b,c\ each of η(x)η(x), η(x−b)η(x-b), η(x−c)η(x-c) is ±1± 1, so the summand equals 88 when x is one of the points counted by N(b,c)N(b,c) and equals 00 otherwise. The three excluded points contribute (1−η(−b))(1−η(−c)),(1+η(b))(1−η(b−c)),(1+η(c))(1−η(c−b)) (1-η(-b) ) (1-η(-c) ), (1+η(b) ) (1-η(b-c) ), (1+η(c) ) (1-η(c-b) ) respectively, and these sum to R(b,c)R(b,c) because η(−u)=η(u)η(-u)=η(u). Hence T=8N(b,c)+R(b,c).T=8N(b,c)+R(b,c). On the other hand, expanding the product gives T=∑x1 T= _x1 +∑xη(x)−∑xη(x−b)−∑xη(x−c) + _xη(x)- _xη(x-b)- _xη(x-c) +∑xη((x−b)(x−c))−∑xη(x(x−b))−∑xη(x(x−c))+S(b,c), + _xη ((x-b)(x-c) )- _xη (x(x-b) )- _xη (x(x-c) )+S(b,c), all sums being over x∈qx _q. The first sum is q, and the three single-character sums vanish because η is nonprincipal. By equation 4 each of the three quadratic sums equals −1-1, so together they contribute −1+1+1=1-1+1+1=1. Therefore T=q+1+S(b,c)T=q+1+S(b,c), and comparing the two expressions for T gives equation 5. It remains to determine the possible values of R(b,c)R(b,c). Write α=η(b)α=η(b), β=η(c)β=η(c) and γ=η(b−c)γ=η(b-c), all of which lie in ±1\± 1\, so that R(b,c)=(1−α)(1−β)+(1+α)(1−γ)+(1+β)(1−γ).R(b,c)=(1-α)(1-β)+(1+α)(1-γ)+(1+β)(1-γ). If γ=1γ=1 then R(b,c)=(1−α)(1−β)R(b,c)=(1-α)(1-β), which is 44 when α=β=−1α=β=-1 and 00 otherwise. If γ=−1γ=-1 then R(b,c)=(1−α)(1−β)+2(1+α)+2(1+β)R(b,c)=(1-α)(1-β)+2(1+α)+2(1+β), which is 88 when α=β=1α=β=1 and 44 in the three remaining cases. This exhausts all possibilities. ∎ C.1.3 Traces of elliptic curves Two facts about elliptic curves over finite fields are needed. The first is Hasse’s bound (Silverman 2009, Chapter V, Theorem 1.1): if E is an elliptic curve over qF_q, then its trace t=q+1−#E(q)t=q+1-\#E(F_q) satisfies |t|≤2q|t|≤ 2 q. The second is the classification of the integers that occur as traces, due to Waterhouse 1969. Theorem C.1.4. Let p be a prime, let r≥1r≥ 1, and let t be an integer with |t|≤2pr/2|t|≤ 2p^r/2. There is an elliptic curve over prF_p^r of trace t if and only if at least one of the following holds: 1. gcd(t,p)=1 (t,p)=1; 2. r is even and t=±2pr/2t=± 2p^r/2; 3. r is even, p≢1(mod3)p ≡ 1 3 and t=±pr/2t=± p^r/2; 4. r is odd, p∈2,3p∈\2,3\ and t=±p(r+1)/2t=± p^(r+1)/2; 5. t=0t=0, and either r is odd or r is even with p≢1(mod4)p ≡ 1 4. The same classification is restated by Schoof 1987, where it is the starting point for counting the isomorphism classes of elliptic curves in a fixed isogeny class. We use it only through the following consequence. Corollary C.1.5. Let p>3p>3 be a prime and let r be odd. If E is an elliptic curve over prF_p^r whose trace t is divisible by p, then t=0t=0. Proof. Of the five cases of Theorem C.1.4, the first is excluded by p|tp t, the second and third by r being odd, and the fourth by p>3p>3. Only the fifth remains, and it gives t=0t=0. ∎ C.1.4 Proof of the main theorem Proof of Theorem C.1.1. Set q=55=3125q=5^5=3125. By vertex-transitivity it suffices to prove that N(b,c)≥377N(b,c)≥ 377 for every pair of distinct b,c∈q∗b,c _q^*, since N(b,c)N(b,c) counts exactly the vertices adjacent to 00 and to neither b nor c. Fix such a pair and consider Eb,c:y2=x(x−b)(x−c).E_b,c: y^2=x(x-b)(x-c). The cubic x(x−b)(x−c)x(x-b)(x-c) has three distinct roots and charq=5≠2charF_q=5≠ 2, so Eb,cE_b,c is nonsingular, that is, an elliptic curve. For each x∈qx _q the number of y∈qy _q with y2=x(x−b)(x−c)y^2=x(x-b)(x-c) is 1+η(x(x−b)(x−c))1+η (x(x-b)(x-c) ), so summing over x and adding the point at infinity gives #Eb,c(q)=q+1+S(b,c).\#E_b,c(F_q)=q+1+S(b,c). Hence the trace of Eb,cE_b,c is t=q+1−#Eb,c(q)=−S(b,c)t=q+1-\#E_b,c(F_q)=-S(b,c), and Hasse’s bound gives |S(b,c)|≤2q=505<112,|S(b,c)|≤ 2 q=50 5<112, so that S(b,c)≥−111S(b,c)≥-111. Combining this with Lemma C.1.3 and R(b,c)≤8R(b,c)≤ 8 yields 8N(b,c)=q+1+S(b,c)−R(b,c)≥3126−111−8=3007,8N(b,c)=q+1+S(b,c)-R(b,c)≥ 3126-111-8=3007, and therefore N(b,c)≥376N(b,c)≥ 376. It remains to exclude the equality case. Suppose N(b,c)=376N(b,c)=376. Then equation 5 reads 3008=3126+S(b,c)−R(b,c),3008=3126+S(b,c)-R(b,c), so S(b,c)=R(b,c)−118S(b,c)=R(b,c)-118. As R(b,c)∈0,4,8R(b,c)∈\0,4,8\ this forces S(b,c)∈−118,−114,−110S(b,c)∈\-118,-114,-110\, and the bound S(b,c)≥−111S(b,c)≥-111 leaves only R(b,c)=8,S(b,c)=−110.R(b,c)=8, S(b,c)=-110. The curve Eb,cE_b,c would then have trace t=−S(b,c)=110t=-S(b,c)=110. But 110110 is a nonzero multiple of 55, and q=55q=5^5 has p=5>3p=5>3 with r=5r=5 odd, so Corollary C.1.5 rules this out. Therefore N(b,c)≠376N(b,c)≠ 376, and the preceding bound gives N(b,c)≥377N(b,c)≥ 377 for every pair of distinct b,c∈q∗b,c _q^*. Hence G3125∈(1,2,377)G_3125 (1,2,377). Finally, (1+22⋅377)2=(1+2754)2=3017+4754>3017+108=3125, (1+2 2· 377 )^2= (1+2 754 )^2=3017+4 754>3017+108=3125, because 272=729<75427^2=729<754. This completes the proof. ∎ References. Ananchuen & Caccetta (1995) W Ananchuen and L Caccetta. 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C.2 A 4-uniform tree that is not 5-good This problem asks whether r-uniform trees are optimal in the Ramsey problem against a complete r-uniform hypergraph, as ordinary trees are against a clique. They are not. The unique 44-uniform tree on seven vertices fails to be 55-good, and the obstruction is a red/blue coloring of K8(4)K_8^(4) whose red edges are exactly the fourteen affine planes of 23F_2^3. C.2.1 Introduction All hypergraphs here are finite and simple. For an r-uniform hypergraph H we write V(H)V(H) and E(H)E(H) for its vertex and edge sets, and Kn(r)K_n^(r) for the complete r-uniform hypergraph on n vertices. The Ramsey number R(G,H,r)R(G,H;r) is the least p such that every red/blue coloring of E(Kp(r))E(K_p^(r)) contains a red copy of G or a blue copy of H. A Berge cycle of length k≥2k≥ 2 in H is an alternating sequence v1,e1,v2,e2,…,vk,ekv_1,e_1,v_2,e_2,…,v_k,e_k of distinct vertices viv_i and distinct edges eie_i with vi,vi+1∈eiv_i,v_i+1∈ e_i for every i, indices read modulo k. An r-uniform tree is a connected r-uniform hypergraph containing no Berge cycle. Equivalently (Budden & Penland 2017, Theorem 2.1), it is a hypergraph that can be assembled starting from a single edge, each subsequent edge meeting the union of the previous ones in exactly one vertex; in particular an r-uniform tree on m vertices has exactly (m−1)/(r−1)(m-1)/(r-1) edges. A weak coloring of H is a coloring of V(H)V(H) in which no edge is monochromatic, the weak chromatic number χw(H) _w(H) is the least number of colors in a weak coloring, and the chromatic surplus s(H)s(H) is the least size of a color class over all weak colorings of H with exactly χw(H) _w(H) colors. Budden & Penland 2017 proved that every connected r-uniform hypergraph G of order m≥rm≥ r satisfies R(G,Kn(r),r)≥(m−1)(χw(Kn(r))−1)+s(Kn(r)),R (G,K_n^(r);r )\ ≥\ (m-1) ( _w(K_n^(r))-1 )+s (K_n^(r) ), (6) and they call G n-good when equality holds; their notation for the chromatic surplus is t(⋅)t(·), and we follow the later s(⋅)s(·) of Budden & Clifton 2022. For an arbitrary r-uniform target H with s(H)≤ms(H)≤ m the same theorem gives R(G,H,r)≥(m−1)(χw(H)−1)+s(H)R(G,H;r)≥(m-1)( _w(H)-1)+s(H), and G is H-good when equality holds; n-goodness is the case H=Kn(r)H=K_n^(r). Here χw(Kn(r))=⌈n/(r−1)⌉ _w(K_n^(r))= n/(r-1) , since a color class of a weak coloring of Kn(r)K_n^(r) is exactly a set of at most r−1r-1 vertices. For r=2r=2, equation 6 reads R(G,Kn)≥(m−1)(n−1)+1R(G,K_n)≥(m-1)(n-1)+1, and Chvátal 1977 proved that every tree attains it: every tree TmT_m on m vertices satisfies R(Tm,Kn)=(m−1)(n−1)+1R(T_m,K_n)=(m-1)(n-1)+1. The term n-good goes back to Burr et al. 1980, and the systematic study of goodness to Burr & Erdös 1983. Budden & Penland 2017 conjectured that hypertrees are optimal in the same sense: If r≥2r≥ 2 and T is any r-uniform tree, then T is n-good [for every n≥rn≥ r]. We disprove this. The evidence behind the conjecture is substantial. Budden & Penland 2017 record the hypertree embedding bound of Loh 2009, namely R(Tm(r),Kn(r),r)≤(m−1)(n−1)r−1+1R (T_m^(r),K_n^(r);r )\ ≤\ (m-1)(n-1)r-1+1 for every r-uniform tree Tm(r)T_m^(r) on m vertices, and deduce that the conjecture holds whenever (r−1)|(n−1)(r-1) (n-1), because the two bounds then agree. For r=3r=3 the divisibility condition says exactly that n is odd, and for the smallest nontrivial tree Budden & Penland 2017 push past it: R(T5(3),Kn(3),3)=2n−2R(T_5^(3),K_n^(3);3)=2n-2 for even n as well, so T5(3)T_5^(3) is n-good for every n≥3n≥ 3. Beyond T5(3)T_5^(3) their bounds for even n leave a range of possible values rather than a single one, and they report finding no counterexample. Budden & Clifton 2022 reduce the conjecture to the residues n≡0(modr−1)n≡ 0 r-1: a tree that is n-good for every multiple n of r−1r-1 is n-good for every n≥rn≥ r. In the asymptotic direction, Boyadzhiyska & Lo 2025 prove that for every r≥3r≥ 3 and every r-uniform hypergraph H with s(H)≤2r−1s(H)≤ 2r-1, all sufficiently long r-uniform loose paths are H-good, the threshold depending on H. Every complete target satisfies the hypothesis, since s(Kn(r))≤r−1s(K_n^(r))≤ r-1, so for each fixed n all sufficiently long loose paths are n-good. A loose path that is not n-good is therefore short, and ours is the shortest nontrivial one. Connected hypergraphs that are not n-good were known, but no tree was among them; Budden & Penland 2017 show that the 33-uniform loose cycle C4(3)C_4^(3), two edges meeting in two vertices, is not 55-good, and record that K4(3)K_4^(3), K5(3)K_5^(3) and K6(3)K_6^(3) are not 44-good and that K5(4)K_5^(4) is not 55-good. Two nearby negative results are of a different kind. The ℓ -paths with ℓ≥2 ≥ 2 that Boyadzhiyska & Lo 2025 rule out are not trees, since consecutive edges meet in ℓ≥2 ≥ 2 vertices and hence span a Berge cycle of length 22. The loose paths they rule out are trees, but the targets there are incomplete hypergraphs H with s(H)>2r−1s(H)>2r-1, outside the hypothesis above. Radziszowski’s dynamic survey describes the subsequent literature as “further results towards the conjecture that all r-uniform trees are n-good” (Radziszowski 1994). (A corrigendum (Budden & Penland 2019) amends Theorem 3.10 of the original paper, on disjoint unions of n-good hypergraphs; Conjecture 6.2 is unaffected.) The counterexample is the smallest 44-uniform tree other than a single edge, equivalently the 44-uniform loose path with two edges, and it is taken against K5(4)K_5^(4), a pair of parameters that the divisibility criterion does not cover. Let T be the 44-uniform hypergraph on the vertex set [7][7] with the two edges e1=1,2,3,4,e2=4,5,6,7.e_1=\1,2,3,4\, e_2=\4,5,6,7\. (7) It is connected, and it has no Berge cycle: a Berge cycle uses at least two distinct edges, so with only e1e_1 and e2e_2 available it would need two distinct vertices in e1∩e2e_1∩ e_2, whereas |e1∩e2|=1|e_1∩ e_2|=1. Hence T is a 44-uniform tree, and it is the unique one of order 77 up to isomorphism, since a 44-uniform tree on 77 vertices has exactly (7−1)/(4−1)=2(7-1)/(4-1)=2 edges and those edges meet in exactly one vertex. Theorem C.2.1. Let T be the 44-uniform tree of equation 7. Then R(T,K5(4),4)=9,R (T,K_5^(4);4 )=9, while the right-hand side of equation 6 equals 88 for G=TG=T, r=4r=4 and n=5n=5. In particular T is not 55-good, and Conjecture 6.2 of Budden & Penland 2017 is false. Remark C.2.2. The 33-uniform case of Conjecture 6.2, which is the bulk of Budden & Penland 2017, remains open. So does their conjecture that any two r-uniform trees of the same order have the same Ramsey number against Kn(r)K_n^(r): T is the only 44-uniform tree of order 77. The divisibility hypothesis of Budden & Penland 2017 is not an artifact of a lossy estimate. Fix n≥rn≥ r, write n−1=q(r−1)+jn-1=q(r-1)+j with 0≤j≤r−20≤ j≤ r-2, and let k=(m−1)/(r−1)k=(m-1)/(r-1) be the number of edges of an r-uniform tree of order m. Then χw(Kn(r))=q+1 _w(K_n^(r))=q+1 and s(Kn(r))=j+1s(K_n^(r))=j+1, so the Loh upper bound and the Burr-type lower bound in equation 6 are (m−1)q+kj+1(m-1)q+kj+1 and (m−1)q+j+1(m-1)q+j+1 respectively, and the window between them has width exactly j(k−1)j(k-1). It closes precisely when (r−1)|(n−1)(r-1) (n-1) or the tree is a single edge. At (r,m,n)=(4,7,5)(r,m,n)=(4,7,5) the window is 8,9\8,9\ and Theorem C.2.1 places the truth at the top of it, whereas at (r,m,n)=(3,5,4)(r,m,n)=(3,5,4) the window is again a single step, 6,7\6,7\, and there the truth is at the bottom: R(T5(3),K4(3),3)=6R(T_5^(3),K_4^(3);3)=6 (Budden & Penland 2017). The lower bound is a single explicit coloring. The key point is that the affine planes of 23F_2^3, regarded as 44-element subsets, meet one another in 00 or 22 points and never in 11, while the two edges of T meet in exactly 11 point; at the same time the planes are numerous enough that every 55 points contain one. The matching upper bound is Loh’s hypertree embedding theorem, which at (r,m,n)=(4,7,5)(r,m,n)=(4,7,5) gives exactly 99. C.2.2 The affine-plane coloring Take the vertex set of K8(4)K_8^(4) to be V=23V=F_2^3. For a∈23∖0a _2^3 \0\ and b∈2b _2 put P(a,b):=x∈23:a⋅x=b,P(a,b):=\x _2^3:\ a· x=b\, where a⋅xa· x denotes the standard bilinear form; such a set is an affine plane, that is, a coset of a two-dimensional subspace. Color a 44-subset of V red if it is an affine plane, and blue otherwise. Figure 7 depicts the two intersection patterns at issue. 11223344556677e1e_1e2e_2the tree T000000100100010010110110001001101101011011111111P(001,0)P(001,0) and P(010,0)P(010,0) Figure 7: On the left, the unique 44-uniform tree T of order 77: its two edges meet in the single vertex 44. On the right, two of the fourteen affine planes of 23F_2^3, drawn as faces of the cube; they meet in the two points 000000 and 100100, marked by the thick segment. Lemma C.2.4 says that two red edges can never meet the way e1e_1 and e2e_2 do. Lemma C.2.3. There are exactly 1414 affine planes in 23F_2^3, and a 44-subset S⊆23S _2^3 is an affine plane if and only if ∑x∈Sx=0 _x∈ Sx=0. Proof. The map a↦P(a,0)a P(a,0) is a bijection from 23∖0F_2^3 \0\ onto the set of two-dimensional subspaces of 23F_2^3, and each such subspace has exactly the two cosets P(a,0)P(a,0) and P(a,1)P(a,1). Hence there are exactly 1414 affine planes, each of size 44. A two-dimensional subspace has the form 0,p,q,p+q\0,p,q,p+q\ and therefore sums to 00, and translating a 44-set by x changes its sum by 4x=04x=0; so every affine plane sums to 00. Conversely, let S=x1,x2,x3,x4S=\x_1,x_2,x_3,x_4\ satisfy ∑ixi=0 _ix_i=0. Then S+x4=x1+x4,x2+x4,x3+x4, 0S+x_4=\x_1+x_4,\,x_2+x_4,\,x_3+x_4,\,0\ consists of 00 together with three distinct nonzero elements p,q,up,q,u whose sum is 00. Thus u=p+qu=p+q, and p,qp,q are distinct nonzero vectors and hence linearly independent, so S+x4=0,p,q,p+qS+x_4=\0,p,q,p+q\ is a two-dimensional subspace. Therefore S is one of its cosets. ∎ C.2.3 The lower bound We now check that the coloring contains neither of the two forbidden configurations. Lemma C.2.4. Any two distinct red edges meet in 00 or 22 vertices. Consequently the coloring contains no red copy of T. Proof. Let P(a,b)≠P(c,d)P(a,b)≠ P(c,d) be red edges. If a=ca=c then b≠db≠ d, and the two sets are disjoint. If a≠ca≠ c then a and c are distinct nonzero vectors, hence linearly independent over 2F_2, so the linear map x↦(a⋅x,c⋅x)x (a· x,\,c· x) from 23F_2^3 to 22F_2^2 is surjective with kernel of dimension 11. Every one of its fibers, in particular P(a,b)∩P(c,d)P(a,b)∩ P(c,d), therefore has exactly 22 elements. A red copy of T would consist of two red edges meeting in exactly one vertex, which the above excludes. ∎ Lemma C.2.5. Every 55-subset of 23F_2^3 contains a red 44-subset. Consequently the coloring contains no blue K5(4)K_5^(4). Proof. Let X⊆23X _2^3 with |X|=5|X|=5, write Y=23∖X=u,v,wY=F_2^3 X=\u,v,w\, and put σ=u+v+wσ=u+v+w. If σ∈Yσ∈ Y, say σ=uσ=u, then v+w=0v+w=0 and hence v=wv=w, a contradiction; so σ∈Xσ∈ X. In each coordinate exactly four of the eight vectors of 23F_2^3 have entry 11, so ∑x∈23x=0 _x _2^3x=0 and therefore ∑x∈Xx=∑y∈Yy=σ. _x∈ Xx= _y∈ Yy=σ. It follows that ∑x∈X∖σx=0 _x∈ X \σ\x=0, so X∖σX \σ\ is red by Lemma C.2.3. A blue K5(4)K_5^(4) would be a 55-set all of whose 44-subsets are blue, which the above excludes. ∎ Lemmas C.2.4 and C.2.5 exhibit a red/blue coloring of K8(4)K_8^(4) with no red T and no blue K5(4)K_5^(4), so R(T,K5(4),4)≥ 9.R (T,K_5^(4);4 )\ ≥\ 9. (8) C.2.4 Proof of the main theorem The remaining input is the hypertree embedding theorem of Loh 2009, which answered a question of Bohman et al. 2010 by removing all dependence on the uniformity r. Theorem C.2.6. Every r-uniform hypergraph with weak chromatic number greater than k contains a copy of every r-uniform tree with k edges. Proof of Theorem C.2.1. For the upper bound, consider any red/blue coloring of E(K9(4))E(K_9^(4)) and let HRH_R be the spanning subhypergraph formed by the red edges. The tree T has exactly 22 edges. If χw(HR)>2 _w(H_R)>2, then Theorem C.2.6 produces a red copy of T. Otherwise fix a weak coloring of HRH_R using at most two colors. Some color class C satisfies |C|≥⌈9/2⌉=5|C|≥ 9/2 =5, and no red edge lies inside a color class, so all 44-subsets of any five vertices of C are blue. That is a blue K5(4)K_5^(4). Hence R(T,K5(4),4)≤9R(T,K_5^(4);4)≤ 9, and with equation 8 we obtain R(T,K5(4),4)=9R(T,K_5^(4);4)=9. It remains to evaluate the right-hand side of equation 6. A weak coloring of K5(4)K_5^(4) is exactly a partition of a 55-set into classes of size at most 33, so two classes are needed and 3+23+2 is the only partition into two such classes. Hence χw(K5(4))=2 _w(K_5^(4))=2 and s(K5(4))=2s(K_5^(4))=2, and since |V(T)|=7|V(T)|=7 the right-hand side of equation 6 equals (7−1)(2−1)+2=8(7-1)(2-1)+2=8. As 9≠89≠ 8, the tree T is not 55-good. ∎ Remark C.2.7. Theorem C.2.1 also settles a value left open in Budden & Penland 2017. Budden & Penland 2017 prove that 2r≤R(T2r−1(r),Kr+1(r),r)≤2r+12r≤ R(T_2r-1^(r),K_r+1^(r);r)≤ 2r+1 for all r≥3r≥ 3, which at r=4r=4 is precisely the window 8≤R(T,K5(4),4)≤98≤ R(T,K_5^(4);4)≤ 9; the value is 99. The failure also propagates upward. By Budden & Clifton 2022, a 44-uniform tree that is 66-good is also 55-good and 44-good, so T is not 66-good either. References. Bohman et al. (2010) Tom Bohman, Alan Frieze, and Dhruv Mubayi. Coloring h-free hypergraphs. Random Structures & Algorithms, 36(1):11–25, 2010. Boyadzhiyska & Lo (2025) Simona Boyadzhiyska and Allan Lo. Ramsey goodness of k-uniform paths, or the lack thereof. European Journal of Combinatorics, 129:104021, 2025. Budden & Clifton (2022) Mark Budden and Justin Clifton. Hypergraph ramsey numbers involving trees, stars, and complete hypergraphs. Integers: Electronic Journal of Combinatorial Number Theory, 22:1, 2022. Budden & Penland (2017) Mark Budden and Andrew Penland. Trees and n-good hypergraphs. arXiv preprint arXiv:1710.05731, 2017. Budden & Penland (2019) Mark Budden and Andrew Penland. Corrigendum: Trees and n-good hypergraphs. Australas. J Comb., 75:171–173, 2019. Burr & Erdös (1983) Stefan A Burr and Paul Erdös. Generalizations of a ramsey-theoretic result of chvátal. Journal of Graph Theory, 7(1):39–51, 1983. Burr et al. (1980) Stefan A Burr, P Erdős, Ralph J Faudree, C Rousseau, and RH Schelp. An extremal problem in generalized ramsey theory. Ars Combinatoria, 10:193–203, 1980. Chvátal (1977) Vasek Chvátal. Tree-complete graph ramsey numbers. Journal of Graph Theory, 1(1):93–93, 1977. Loh (2009) Po-Shen Loh. A note on embedding hypertrees. arXiv preprint arXiv:0901.2988, 2009. Radziszowski (1994) Stanisław P. Radziszowski. Small Ramsey numbers. The Electronic Journal of Combinatorics, 1994. doi: 10.37236/21. URL https://w.combinatorics.org/ojs/index.php/eljc/article/view/DS1. Dynamic Survey DS1, revision 18, 24 April 2026. C.3 Maximum versus average independent set size in triangle-free graphs This problem asks by how much the largest independent set of a triangle-free graph exceeds a typical one. Davies et al. 2018 conjectured that a factor 2−od(1)2-o_d(1) is forced once the minimum degree d is large. We disprove this. The Cartesian products C5□Km,mC_5\, \,K_m,m are triangle-free and (m+2)(m+2)-regular, and for them the ratio of the maximum to the average size of an independent set tends to 24/13=1.846…24/13=1.846…, which is bounded away from 22. The mechanism is that the two sides of Km,mK_m,m must draw their independent sets from disjoint parts of the pentagon. C.3.1 Introduction For a finite graph G let ℐ(G)I(G) be its family of independent sets and α(G)=maxI∈ℐ(G)|I|α(G)= _I (G)|I| its independence number. The hard-core model on G at fugacity λ>0λ>0 is the probability distribution on ℐ(G)I(G) giving each I mass proportional to λ|I|λ^|I|, and α¯G(λ)=∑I∈ℐ(G)|I|λ|I|∑I∈ℐ(G)λ|I| α_G(λ)= _I (G)|I|\,λ^|I| _I (G)λ^|I| is the expected size of a set drawn from it. At λ=1λ=1 the distribution is uniform on ℐ(G)I(G), so α¯G(1) α_G(1) is the average size of an independent set of G; it lies between 00 and α(G)α(G) and is not normalized by |V(G)||V(G)|. Davies et al. 2018 proved that a triangle-free graph G on n vertices with Δ(G)≤d (G)≤ d satisfies α¯G(1)≥(1+od(1))logdn α_G(1)≥(1+o_d(1)) dd\,n, so that the average independent set is already as large as the lower bound Shearer’s theorem (Shearer 1983) gives for the maximum one; in particular this gives a second proof of R(3,k)≤(1+o(1))k2/logkR(3,k)≤(1+o(1))k^2/ k. In Section 5 of the same paper they ask by how much the maximum exceeds the average and record four conjectures: three on triangle-free graphs and a fourth, Conjecture 4, for KrK_r-free graphs. The second of them, Davies et al. 2018, reads as follows: For every triangle-free graph G of minimum degree d, α(G)/α¯G(1)≥2−od(1)α(G)/ α_G(1)≥ 2-o_d(1). Davies et al. 2018 deduce from it that R(3,k)≤(12+o(1))k2/logkR(3,k)≤( 12+o(1))k^2/ k, halving the constant in Shearer’s bound, which is still the best known. The general lower bound available for all graphs is Davies et al. 2018, which states that α(G)/α¯G(λ)≥1+α(G)/(λn)α(G)/ α_G(λ)≥ 1+α(G)/(λ n) for every graph G on n vertices, with equality for a disjoint union of copies of a single KrK_r. That bound degenerates as soon as α(G)=o(n)α(G)=o(n). The constant 4/34/3 of Conjecture 1 is the exact value of α(G)/α¯G(1)α(G)/ α_G(1) at G=K3G=K_3, and it also falls below 197136/137585=1.43283…197136/137585=1.43283…, the smallest ratio Davies et al. 2018 report, attained by the cyclic triangle-free graph witnessing R(3,9)≥36R(3,9)≥ 36 (Grinstead & Roberts 1982). The evidence offered for Conjecture 2 is the expectation that graphs produced by the triangle-free process have ratio tending to 22. The conjecture remained open: Davies & Kang 2025 record it as Conjecture B in the open problems section of their survey of the hard-core model in graph theory, and Morris 2026 restates it in his ICM survey of Ramsey theory, noting that any lower bound better than 1+o(1)1+o(1) would be a very significant breakthrough. The one route toward it proposed in Davies et al. 2018 is the identity α(G)=α¯G(1)+∫1∞Varλ(|I|)λλ,α(G)= α_G(1)+ _1^∞ Var_λ(|I|)λ\,dλ, which reduces the conjecture to lower bounds on the variance of the hard-core model; Davies et al. 2025 prove such bounds, but only for fugacities that are small in terms of the number of vertices. In a neighboring circle of hard-core conjectures, Cambie & Jooken 2023 disproved five conjectures on the extremal graphs for the occupancy fraction and the independence polynomial among regular graphs of given girth, by computer search over small graphs. Theorem C.3.1. For m≥1m≥ 1 let Gm=C5□Km,mG_m=C_5\, \,K_m,m. Then GmG_m is triangle-free and (m+2)(m+2)-regular on 10m10m vertices, α(Gm)=4mα(G_m)=4m, and α¯Gm(1)=136m(1+O((2/3)m)),so thatα(Gm)α¯Gm(1)⟶2413=1.846…<2. α_G_m(1)= 136\,m (1+O ((2/3)^m ) ), that α(G_m) α_G_m(1) 2413=1.846…<2. Corollary C.3.2. Conjecture 2 of Davies et al. 2018 is false. Indeed GmG_m is triangle-free of minimum degree d=m+2→∞d=m+2→∞ while α(Gm)/α¯Gm(1)→24/13α(G_m)/ α_G_m(1)→ 24/13, so no bound of the form 2−od(1)2-o_d(1) can hold. Since a triangle-free graph is KrK_r-free for every r≥3r≥ 3, the same graphs refute the minimum-degree assertion of Davies et al. 2018 for every fixed r≥3r≥ 3. The product structure is essential: Km,mK_m,m alone is triangle-free and m-regular with α=mα=m, |ℐ(Km,m)|=2m+1−1|I(K_m,m)|=2^m+1-1 and ∑I|I|=m2m _I|I|=m2^m, so its ratio is exactly 2−2−m2-2^-m. Conjecture 1 of Davies et al. 2018, that α(G)/α¯G(1)≥4/3α(G)/ α_G(1)≥ 4/3 for every triangle-free graph G, is untouched by these examples, since both 24/1324/13 and the value 32/1932/19 obtained below exceed 4/34/3; so is the first assertion of Conjecture 4, which asks only for 1+1/r1+1/r in the KrK_r-free case. Conjecture 3 of Davies et al. 2018, the version at general fugacity, is also untouched: we compute only at λ=1λ=1, whereas that conjecture is free to choose λ small, and by the same deduction it still implies R(3,k)≤(12+o(1))k2/logkR(3,k)≤( 12+o(1))k^2/ k. Finally, α(Gm)/|V(Gm)|=2/5α(G_m)/|V(G_m)|=2/5, so these graphs lie in the range where the general bound of Davies et al. 2018 already forces the ratio to be at least 7/57/5; they say nothing about triangle-free graphs with α(G)=o(|V(G)|)α(G)=o(|V(G)|), which is the range relevant to R(3,k)R(3,k). We now summarize the computation. An independent set of C5□Km,mC_5\, \,K_m,m is a family of independent sets of C5C_5, one over each vertex of Km,mK_m,m, subject only to the requirement that the union of those over one side be disjoint from the union of those over the other. Classifying such a family by its pair of unions and inverting over the Boolean lattice 2ℤ52^Z_5 writes the independence polynomial of GmG_m as a signed sum of 353^5 terms (FXFY)m(F_XF_Y)^m indexed by the disjoint pairs X,Y⊆ℤ5X,Y _5, where FXF_X is the independence polynomial of the subgraph of C5C_5 induced on X. The key point is that FX(1)FY(1)F_X(1)F_Y(1) is maximized only when one of X,YX,Y is a nonadjacent pair of C5C_5 and the other is its complement. The mean 32+23=136 32+ 23= 136 attached to that configuration is the average number of chosen vertices per pair of opposite fibers, which gives α¯Gm(1)≈136m α_G_m(1)≈ 136m. C.3.2 The construction In the Cartesian product G□HG\, \,H the vertex set is V(G)×V(H)V(G)× V(H), and (g,h)(g,h) is adjacent to (g′,h′)(g ,h ) exactly when g=g′g=g and hh′∈E(H)h ∈ E(H), or h=h′h=h and gg′∈E(G)g ∈ E(G). Degrees add, so Gm=C5□Km,mG_m=C_5\, \,K_m,m is (m+2)(m+2)-regular on 5⋅2m=10m5· 2m=10m vertices, and in particular δ(Gm)=m+2δ(G_m)=m+2. Lemma C.3.3. The Cartesian product of two triangle-free graphs is triangle-free. In particular GmG_m is triangle-free. Proof. Every edge of G□HG\, \,H changes exactly one coordinate. Let u,v,wu,v,w span a triangle. By the pigeonhole principle two of its three edges change the same coordinate, say the first, and these two edges share a vertex, say uvuv and vwvw. Then u,v,wu,v,w all have the same second coordinate, so the third edge uwuw also changes only the first coordinate. Hence the first coordinates of u,v,wu,v,w are pairwise distinct and pairwise adjacent, giving a triangle in G. The same argument with the coordinates exchanged gives a triangle in H when the repeated coordinate is the second. Since C5C_5 and Km,mK_m,m are triangle-free, so is GmG_m. ∎ Throughout, ℤ5=0,1,2,3,4Z_5=\0,1,2,3,4\ is the vertex set of C5C_5, with i adjacent to i±1i± 1, and L and R are the two sides of Km,mK_m,m, each of size m. For X⊆ℤ5X _5 we write ℐ(X)I(X) for the family of independent sets of the induced subgraph C5[X]C_5[X] and put FX(y)=∑S∈ℐ(X)y|S|,f(X)=FX(1)=|ℐ(X)|,F_X(y)= _S (X)y^|S|, f(X)=F_X(1)=|I(X)|, so that FXF_X is the independence polynomial of C5[X]C_5[X]. Observe that f is monotone: X⊆X′X X implies f(X)≤f(X′)f(X)≤ f(X ). The fiber ℤ5×xZ_5×\x\ over a vertex x of Km,mK_m,m induces a copy of C5C_5, and the two fibers over adjacent x,x′x,x are joined by a perfect matching that preserves the ℤ5Z_5-coordinate, as in Figure 8. Hence an independent set I of GmG_m is the same thing as a family (Sx)x∈L∪R(S_x)_x∈ L∪ R with Sx∈ℐ(ℤ5)for every x,Sx∩Sx′=∅whenever xx′∈E(Km,m),S_x (Z_5)\ for every x, S_x∩ S_x = \ whenever x ∈ E(K_m,m), and then |I|=∑x|Sx||I|= _x|S_x|. Since Km,mK_m,m is complete bipartite, the disjointness constraints say precisely that (⋃x∈LSx)∩(⋃x∈RSx)=∅. ( _x∈ LS_x )∩ ( _x∈ RS_x )= . (9) 00112233440011223344x∈Lx∈ Lx′∈Rx ∈ R Figure 8: Two fibers of Gm=C5□Km,mG_m=C_5\, \,K_m,m, over adjacent vertices x∈Lx∈ L and x′∈Rx ∈ R. Each fiber induces a copy of C5C_5, and the gray matching between them preserves the ℤ5Z_5-coordinate, so the independent sets chosen in fibers on opposite sides must be disjoint. The gray halos mark a dominant splitting from Lemma C.3.6: every fiber over L takes its independent set inside the nonadjacent pair X=0,2X=\0,2\ and every fiber over R inside the complement Y=1,3,4Y=\1,3,4\, and this is the splitting of ℤ5Z_5 that maximizes f(X)f(Y)f(X)f(Y). The filled vertices show one admissible choice inside them, namely 0,2\0,2\ over x and 1,3\1,3\ over x′x . Lemma C.3.4. α(Gm)=4mα(G_m)=4m. Proof. Each SxS_x is an independent set of C5C_5, hence |Sx|≤2|S_x|≤ 2, and summing over the 2m2m fibers gives α(Gm)≤4mα(G_m)≤ 4m. For the lower bound take Sx=0,2S_x=\0,2\ for every x∈Lx∈ L and Sx=1,3S_x=\1,3\ for every x∈Rx∈ R. Both sets are independent in C5C_5 and they are disjoint, so equation 9 holds and the resulting independent set has size 4m4m. ∎ C.3.3 Exact counting by Möbius inversion Let Zm(y)=∑I∈ℐ(Gm)y|I|Z_m(y)= _I (G_m)y^|I| be the independence polynomial of GmG_m, so that |ℐ(Gm)|=Zm(1)|I(G_m)|=Z_m(1) and α¯Gm(1)=Zm′(1)/Zm(1) α_G_m(1)=Z_m (1)/Z_m(1). One is tempted to sum (FX(y)FY(y))m (F_X(y)F_Y(y) )^m over the disjoint pairs (X,Y)(X,Y), reading X and Y as the two unions in equation 9. This overcounts: the quantity (FX(y)FY(y))m (F_X(y)F_Y(y) )^m records the families with Sx⊆XS_x X for x∈Lx∈ L and Sx⊆YS_x Y for x∈Rx∈ R, so a single independent set is counted once for every pair (X,Y)(X,Y) whose two parts contain its two unions. The remedy is to force the unions to be attained exactly. For X⊆ℤ5X _5 define gX(y)=∑(S1,…,Sm)∈ℐ(ℤ5)mS1∪⋯∪Sm=Xy|S1|+⋯+|Sm|,g_X(y)= _ subarrayc(S_1,…,S_m) (Z_5)^m\\ S_1∪…∪ S_m=X subarrayy^|S_1|+…+|S_m|, the generating function of the m-tuples of independent sets of C5C_5 whose union is exactly X. A tuple has union contained in X if and only if every SiS_i lies in ℐ(X)I(X), so ∑X′⊆XgX′(y)=FX(y)m. _X Xg_X (y)=F_X(y)^m. Möbius inversion in the Boolean lattice 2ℤ52^Z_5 therefore gives gX(y)=∑X′⊆X(−1)|X∖X′|FX′(y)m.g_X(y)= _X X(-1)^|X X |F_X (y)^m. (10) Classifying an independent set of GmG_m by the ordered pair of unions in equation 9 partitions ℐ(Gm)I(G_m), and yields the exact identity Zm(y)=∑X,Y⊆ℤ5X∩Y=∅gX(y)gY(y).Z_m(y)= _ subarraycX,Y _5\\ X∩ Y= subarrayg_X(y)\,g_Y(y). (11) Substituting equation 10 into equation 11 collapses to a sum over the same index set with explicit signs. Lemma C.3.5. For every m≥1m≥ 1, Zm(y)=∑X,Y⊆ℤ5X∩Y=∅(−1) 5−|X|−|Y|(FX(y)FY(y))m.Z_m(y)= _ subarraycX,Y _5\\ X∩ Y= subarray(-1)^\,5-|X|-|Y| (F_X(y)F_Y(y) )^m. Proof. If X∩Y=∅X∩ Y= and X′⊆X X, Y′⊆Y Y, then X′∩Y′=∅X ∩ Y = , so expanding equation 10 in equation 11 produces a linear combination of the terms (FX′FY′)m (F_X F_Y )^m indexed by disjoint pairs (X′,Y′)(X ,Y ). Fix such a pair and set W=ℤ5∖(X′∪Y′)W=Z_5 (X ∪ Y ). The pairs (X,Y)(X,Y) contributing to it are exactly X=X′∪AX=X ∪ A and Y=Y′∪BY=Y ∪ B with A,BA,B disjoint subsets of W, and each contributes the sign (−1)|A|+|B|(-1)^|A|+|B|. Assigning each w∈Ww∈ W independently to A, to B, or to neither, with respective weights −1,−1,+1-1,-1,+1, the coefficient equals ∑A,B⊆WA∩B=∅(−1)|A|+|B|=∏w∈W(1−1−1)=(−1)|W|=(−1) 5−|X′|−|Y′|.∎ _ subarraycA,B W\\ A∩ B= subarray(-1)^|A|+|B|= _w∈ W(1-1-1)=(-1)^|W|=(-1)^\,5-|X |-|Y |. C.3.4 The dominant pairs By Lemma C.3.5 the exponential growth of Zm(1)Z_m(1) is governed by maxf(X)f(Y) f(X)f(Y) over disjoint pairs. We determine both the maximum and the next value, the latter being what controls the error term. Lemma C.3.6. Let X,Y⊆ℤ5X,Y _5 be disjoint. Then f(X)f(Y)≤24f(X)f(Y)≤ 24, with equality exactly when one of X,YX,Y is a pair of nonadjacent vertices of C5C_5 and the other is its complement, which happens for 1010 ordered pairs. Moreover, if f(X)f(Y)≠24f(X)f(Y)≠ 24 then f(X)f(Y)≤16f(X)f(Y)≤ 16. Proof. A direct count gives the value of f on every induced subgraph of C5C_5: it is 11 on ∅ , 22 on a single vertex, 33 on an edge and 44 on a nonedge, 55 on P3P_3 and 66 on K2∪K1K_2∪ K_1, 88 on P4P_4, and 1111 on C5C_5 itself. Since f is monotone and X∩Y=∅X∩ Y= , replacing Y by ℤ5∖XZ_5 X can only increase f(X)f(Y)f(X)f(Y). It therefore suffices to tabulate the complementary pairs (X,ℤ5∖X)(X,Z_5 X), up to the rotational symmetry of C5C_5 and up to swapping the two parts: |X|C5[X]C5[ℤ5∖X]f(X)f(ℤ5∖X)0∅C51⋅11=111K1P42⋅8=162,Xan edgeK2P33⋅5=152,Xa nonedgeK2¯K2∪K14⋅6=24 array[]c|c|c|c|X|&C_5[X]&C_5[Z_5 X]&f(X)f(Z_5 X)\\ 0& &C_5&1· 11=11\\ 1&K_1&P_4&2· 8=16\\ 2,\ X\ an edge&K_2&P_3&3· 5=15\\ 2,\ X\ a nonedge& K_2&K_2∪ K_1&4· 6=24 array The cases |X|≥3|X|≥ 3 are obtained by swapping the two parts. Hence maxf(X)f(Y)=24 f(X)f(Y)=24 over all disjoint pairs, attained on complementary pairs only in the stated configuration; there are 55 nonadjacent pairs in C5C_5, giving 1010 ordered pairs. Suppose now f(X)f(Y)>16f(X)f(Y)>16. Enlarging Y to ℤ5∖XZ_5 X gives f(X)f(ℤ5∖X)>16f(X)f(Z_5 X)>16, so by the table X is either a nonadjacent pair or the complement of one. If X=a,a+2X=\a,a+2\ is the nonadjacent pair then f(X)=4f(X)=4, so f(Y)>4f(Y)>4; but Y is contained in ℤ5∖XZ_5 X, an edge plus an isolated vertex, whose proper subsets all have f≤4f≤ 4, so Y=ℤ5∖XY=Z_5 X and f(X)f(Y)=24f(X)f(Y)=24. If instead f(X)=6f(X)=6 then f(Y)>16/6f(Y)>16/6 with Y contained in a nonadjacent pair, whence f(Y)∈1,2,4f(Y)∈\1,2,4\ and only f(Y)=4f(Y)=4 survives, again giving 2424. Therefore every value other than 2424 is at most 1616. ∎ For a dominant pair, say X=0,2X=\0,2\ and Y=1,3,4Y=\1,3,4\, the induced subgraphs are K2¯ K_2 and K2∪K1K_2∪ K_1, so FX(y)=(1+y)2,FY(y)=(1+y)(1+2y),P(y):=FX(y)FY(y)=(1+y)3(1+2y).F_X(y)=(1+y)^2, F_Y(y)=(1+y)(1+2y), P(y):=F_X(y)F_Y(y)=(1+y)^3(1+2y). Thus P(1)=8⋅3=24P(1)=8· 3=24 and P′(1)P(1)=31+y|y=1+21+2y|y=1=32+23=136. P (1)P(1)= . 31+y |_y=1+ . 21+2y |_y=1= 32+ 23= 136. (12) All five rotations of the pair (X,Y)(X,Y), in both orders, give the same polynomial P. Since P′(1)/P(1)P (1)/P(1) is the sum of the mean sizes of a uniformly random element of ℐ(X)I(X) and of ℐ(Y)I(Y), the constant 13/613/6 has a per-vertex reading: a vertex of L contributes on average 11, the mean of |S||S| over the four independent subsets of a nonadjacent pair, and a vertex of R contributes 7/67/6, the mean of |S||S| over the six independent subsets of an edge plus an isolated vertex. C.3.5 Proof of the main theorem Proof of Theorem C.3.1. The graph-theoretic assertions are Lemmas C.3.3 and C.3.4 together with the degree count above. Split the sum of Lemma C.3.5 into the 1010 dominant pairs, each of which contributes +P(y)m+P(y)^m by Lemma C.3.6 and the sign (−1)5−2−3=+1(-1)^5-2-3=+1, and a remainder: Zm(y)=10P(y)m+Em(y),Em(y)=∑(X,Y)disjointnot dominant(−1)5−|X|−|Y|(FX(y)FY(y))m.Z_m(y)=10\,P(y)^m+E_m(y), E_m(y)= _ subarrayc(X,Y)\ disjoint\\ not dominant subarray(-1)^5-|X|-|Y| (F_X(y)F_Y(y) )^m. Assigning each element of ℤ5Z_5 to X, to Y, or to neither shows that there are 35=2433^5=243 ordered disjoint pairs in all, and by Lemma C.3.6 every nondominant one has FX(1)FY(1)≤16F_X(1)F_Y(1)≤ 16. Hence |Em(1)|≤243⋅16m.|E_m(1)|≤ 243· 16^m. For the derivative, each nondominant term satisfies |dy(FX(y)FY(y))m|y=1=m(FX(1)FY(1))m⋅(FXFY)′(1)FX(1)FY(1)≤4m⋅16m, | ddy (F_X(y)F_Y(y) )^m |_y=1=m (F_X(1)F_Y(1) )^m· (F_XF_Y) (1)F_X(1)F_Y(1)≤ 4m· 16^m, because (FXFY)′(1)/(FXFY)(1)(F_XF_Y) (1)/(F_XF_Y)(1) is a sum of two mean independent set sizes in C5C_5, each at most 22. Hence |Em′(1)|≤972m 16m|E_m (1)|≤ 972\,m\,16^m. Using P(1)=24P(1)=24 and equation 12, Zm(1)=10⋅24m(1+O((2/3)m)),Zm′(1)=136m⋅10⋅24m(1+O((2/3)m)),Z_m(1)=10· 24^m (1+O ((2/3)^m ) ), Z_m (1)= 136\,m· 10· 24^m (1+O ((2/3)^m ) ), since 972m 16m972\,m\,16^m divided by 136m⋅10⋅24m 136m· 10· 24^m is O((2/3)m)O ((2/3)^m ). Dividing, α¯Gm(1)=Zm′(1)Zm(1)=136m(1+O((2/3)m)), α_G_m(1)= Z_m (1)Z_m(1)= 136\,m (1+O ((2/3)^m ) ), and with α(Gm)=4mα(G_m)=4m the ratio tends to 4/(13/6)=24/134/(13/6)=24/13, completing the proof. ∎ Remark C.3.7. The exact values, from Lemma C.3.5, are m|ℐ(Gm)|∑I|I|α¯Gm(1)α(Gm)/α¯Gm(1)181200200/8181/50=1.6200…23561162605420/11872374/1355=1.7520…3113541746850248950/37847227082/124475=1.8243… array[]c|c|c|c|cm&|I(G_m)|& _I|I|& α_G_m(1)&α(G_m)/ α_G_m(1)\\ 1&81&200&200/81&81/50=1.6200…\\ 2&3561&16260&5420/1187&2374/1355=1.7520…\\ 3&113541&746850&248950/37847&227082/124475=1.8243… array The ratio is not monotone in m: it climbs above its limit 24/13=1.8461…24/13=1.8461…, and its largest value 426826898/228757635=1.8658…426826898/228757635=1.8658… is attained at m=6m=6. It never exceeds 1.871.87, since the bounds on Em(1)E_m(1) and Em′(1)E_m (1) in the proof of Theorem C.3.1 give α(Gm)α¯Gm(1)≤4(10+243(2/3)m)653−972(2/3)m<1.864(m≥22), α(G_m) α_G_m(1)≤ 4 (10+243(2/3)^m ) 653-972(2/3)^m<1.864 (m≥ 22), and the remaining m≤21m≤ 21 are evaluated directly from Lemma C.3.5. Remark C.3.8. The same computation applies with C5C_5 replaced by any triangle-free graph H, the sum in Lemma C.3.5 then running over the disjoint pairs of subsets of V(H)V(H) with sign (−1)|V(H)|−|X|−|Y|(-1)^|V(H)|-|X|-|Y|. Take for H the circulant C13(1,5)C_13(1,5) on ℤ13Z_13, in which i is joined to i±1i± 1 and i±5i± 5; it is 44-regular and triangle-free with α(H)=4α(H)=4, so it witnesses R(3,5)≥14R(3,5)≥ 14, and the matching upper bound is due to Greenwood & Gleason 1955. Then C13(1,5)□Km,mC_13(1,5)\, \,K_m,m is triangle-free and (m+4)(m+4)-regular, and it has independence number 8m8m because H contains two disjoint independent 44-sets. Over disjoint pairs X,Y⊆ℤ13X,Y _13 the maximum of f(X)f(Y)f(X)f(Y) is 1728=36⋅481728=36· 48, attained at exactly 5252 ordered pairs, all complementary with |X|,|Y|=6,7\|X|,|Y|\=\6,7\ and hence of sign +1+1; the next value is 14401440, and every dominant pair has mean 19/419/4 in place of 13/613/6. The argument above then gives α¯⟶819/4=3219=1.6842… α α 819/4= 3219=1.6842… with relative error O((5/6)m)O ((5/6)^m ). Replacing C5C_5 by [q][q] with clique fibers for q≥2q≥ 2, that is taking the graph on (L∪R)×[q](L∪ R)×[q] in which each fiber x×[q]\x\×[q] is a KqK_q and (x,c)(x,c) is joined to (x′,c)(x ,c) for all xx′∈E(Km,m)x ∈ E(K_m,m), gives Kq+1K_q+1-free graphs of minimum degree m+q−1m+q-1 whose ratio tends to ρq=2/(a+1+b+1) _q=2/ ( aa+1+ bb+1 ) with a=⌊q/2⌋a= q/2 and b=⌈q/2⌉b= q/2 ; here q=2q=2 gives the triangle-free graph K2□Km,mK_2\, \,K_m,m and the value 22, while q=3q=3 gives 12/712/7. Finally, for a disjoint union independence numbers add and independence polynomials multiply, so α¯ α adds as well and the ratio of a disjoint union of t copies of GmG_m equals that of GmG_m; consequently, for each fixed minimum degree d=m+2d=m+2 that same ratio is realized by triangle-free graphs on arbitrarily many vertices, and the counterexample is not an artifact of the number of vertices being tied to the degree. References. Cambie & Jooken (2023) Stijn Cambie and Jorik Jooken. Counterexamples to conjectures on the occupancy fraction of graphs. arXiv preprint arXiv:2311.05542, 2023. Davies & Kang (2025) Ewan Davies and Ross J Kang. The hard-core model in graph theory. arXiv preprint arXiv:2501.03379, 2025. Davies et al. (2018) Ewan Davies, Matthew Jenssen, Will Perkins, and Barnaby Roberts. On the average size of independent sets in triangle-free graphs. Proceedings of the American Mathematical Society, 146(1):111–124, 2018. Davies et al. (2025) Ewan Davies, Juspreet Singh Sandhu, and Brian Tan. On expectations and variances in the hard-core model on bounded degree graphs. arXiv preprint arXiv:2505.13396, 2025. Greenwood & Gleason (1955) Robert E Greenwood and Andrew Mattei Gleason. Combinatorial relations and chromatic graphs. Canadian Journal of Mathematics, 7:1–7, 1955. Grinstead & Roberts (1982) Charles M Grinstead and Sam M Roberts. On the ramsey numbers r (3, 8) and r (3, 9). Journal of Combinatorial Theory, Series B, 33(1):27–51, 1982. Morris (2026) Robert Morris. Some recent results in ramsey theory. In International Congress of Mathematicians 2026, p. 210–239. SIAM, 2026. Shearer (1983) James B Shearer. A note on the independence number of triangle-free graphs. Discrete Mathematics, 46(1):83–87, 1983. C.4 Sets with no large divisor difference This problem asks how large a set A⊆[n]A [n] can be if no difference b−a≥tb-a≥ t between two of its elements divides b. The odd numbers form such a set of size ⌈n/2⌉ n/2 , and Erdős asked whether |A|≤(12+ot(1))n A ≤ ( 12+o_t(1) )n must hold. We prove that it does, with the explicit error term 3n/loglogn3n/ n. The proof weights each integer by the number of its prime factors drawn from the odd primes in (t,n1/3](t,n^1/3], compares the even elements of A with the odd integers omitted from it through the injection a↦a−pa a-p, and converts the resulting weighted inequality into a bound on |A| A by Cauchy–Schwarz. C.4.1 Introduction Erdős asked, in a letter to Ruzsa written around 1980, how dense a set of integers can be if no sufficiently large difference between two of its elements divides one of them. The question is recorded in the miscellany of Erdős problems of Guy 1983 and in Ruzsa’s survey of Erdős’s work on the integers (Ruzsa 1999), and is Problem 635 on the Erdős problems website (Bloom 2026), where it reads as follows. Let t≥1t≥ 1 and A⊆1,…,NA \1,…,N\ be such that whenever a,b∈Aa,b∈ A with b−a≥tb-a≥ t we have b−a∤b-a b. How large can |A| A be? Is it true that |A|≤(12+ot(1))N A ≤ ( 12+o_t(1) )N? We write n throughout for the integer called N there. We answer the second question affirmatively, with an explicit rate. Fix an integer t≥1t≥ 1. Call a set A⊆[n]A [n] t-admissible if no two elements a<ba<b of A satisfy b−a≥tb-a≥ t and b−a|b-a b, and write F(n,t)F(n;t) for the largest size of a t-admissible subset of [n][n]. Since b=a+(b−a)b=a+(b-a), the divisibility b−a|b-a b is equivalent to b−a|ab-a a, so the forbidden configuration is symmetric in the two elements; we call such a difference a divisor difference of the pair. Enlarging t constrains fewer pairs, so F(n,t)F(n;t) is nondecreasing in t. Two cases are immediate. A 11-admissible set contains no two consecutive integers, since a difference of 11 is at least t=1t=1 and divides everything, so F(n,1)≤⌈n/2⌉F(n;1)≤ n/2 . The odd numbers in [n][n] are 11-admissible, because the difference of two odd numbers is even and an even number cannot divide an odd one, so in fact F(n,1)=⌈n/2⌉F(n;1)= n/2 , and F(n,t)≥⌈n/2⌉F(n;t)≥ n/2 for every t. Once n≥2n≥ 2, the odd numbers stop being optimal as soon as t≥2t≥ 2. Erdős observed that the set A=m≤n:m odd∪2k≤n:k oddA=\m≤ n:m odd\∪\2^k≤ n:k odd\ is 22-admissible (Bloom 2026). Indeed, pairs of odd elements are handled by the previous paragraph. Two powers 2k<2j2^k<2^j have j−k≥2j-k≥ 2, since j and k are distinct odd numbers, so their difference 2k(2j−k−1)2^k(2^j-k-1) has odd part 2j−k−1>12^j-k-1>1 and therefore does not divide 2j2^j. If exactly one of a<ba<b is an added power, then b−ab-a is odd, while the equivalent divisibilities b−a|ab-a a and b−a|b-a b make b−ab-a a divisor of that power of two, forcing b−a=1<2b-a=1<2. Since the two sets above are disjoint, this gives F(n,t)≥⌈n2⌉+log2n−12for every t≥2,F(n;t)\ ≥\ n2 + _2n-12 every t≥ 2, so the inequality asked for cannot hold with ot(1)o_t(1) replaced by 00. The second question was answered affirmatively in January 2026 by Liam Price with ChatGPT-5.2, as recorded on the problem’s page (Bloom 2026); the first question, which asks for the exact value of F(n,t)F(n;t), remains open. Tao observed that an affirmative answer also follows quickly from an inequality of Elliott 2012, which bounds a weighted mean square, over the small primes p, of the difference between the average of an arbitrary function on an interval I and its average along the multiples of p in I, by the mean square of that function on I. Elliott’s inequality can be substituted for Lemma C.4.4 below. Tao also pointed out that the graph in which a and b are joined whenever b−ab-a divides b, once b−ab-a is restricted to primes or almost primes, is closely related to the divisibility graphs through which pair correlations of multiplicative functions are studied (Matomäki et al. 2016; Helfgott & Radziwił 2021); that literature is concerned with connectivity and expansion rather than with independent sets, which is what Erdős’s question asks about (Bloom 2026). Theorem C.4.1. For every integer t≥1t≥ 1 there is an n0(t)n_0(t) such that ⌈n2⌉≤F(n,t)≤n2+3nloglogn n2 \ ≤\ F(n;t)\ ≤\ n2+ 3n n for all n≥n0(t)n≥ n_0(t). In particular F(n,t)=(12+ot(1))nF(n;t)= ( 12+o_t(1) )n for each fixed t. Remark C.4.2. The proof of Theorem C.4.1 given below was found independently of, and later than, the answer recorded on the problem’s page (Bloom 2026), which has priority. The inequality behind Tao’s alternative route, due to Elliott 2012, is contemporaneous with Erdős’s question, and since the deduction from it is short Tao notes that the statement may already be present in the literature (Bloom 2026). Remark C.4.3. Erdős’s construction above gives F(n,t)≥n/2+Ω(logn)F(n;t)≥ n/2+ ( n) for t≥2t≥ 2, whereas the error term proved here is of size n/loglogn/ n. Tao has noted that arguments of Elliott’s type give an error term Ot(n/loglogn)O_t(n/ n), still far above the Ω(logn) ( n) of the construction (Bloom 2026). The upper bound rests on a single injection. Let p be an odd prime with p>tp>t. If a is an even multiple of p, then a−pa-p is an odd multiple of p at distance p≥tp≥ t from a, and p divides a; so a and a−pa-p cannot both lie in a t-admissible set. Summing this over a set P of odd primes larger than t weights each even element of a t-admissible set A, and each odd element excluded from A, by its number of prime factors drawn from P. That weight has mean κ=∑p∈P1/pκ= _p∈ P1/p on each parity class, and its total square deviation there is at most nκ/2+O(|P|2)nκ/2+O( P ^2), so Cauchy–Schwarz converts the weighted inequality into a bound of size n/κn/ κ for the excess of even elements of A over missing odd elements. Taking P to be the odd primes in (t,n1/3](t,n^1/3] makes κ as large as loglogn n. C.4.2 A second moment estimate The following elementary estimate will be used. Lemma C.4.4. Let P be a finite set of odd primes, and put κ=∑p∈P1p,ωP(m)=#p∈P:p∣m.κ= _p∈ P 1p, _P(m)=\#\p∈ P:p m\. Then for every n≥1n≥ 1, ∑m≤nm even(ωP(m)−κ)2≤nκ2+2|P|2and∑m≤nm odd(ωP(m)−κ)2≤nκ2+2|P|2. _ subarraycm≤ n\\ m even subarray ( _P(m)-κ )^2≤ nκ2+2 P ^2 _ subarraycm≤ n\\ m odd subarray ( _P(m)-κ )^2≤ nκ2+2 P ^2. Proof. Let ℳM be either the set of even integers in [n][n] or the set of odd ones, and for an odd squarefree d put N(d)=#m∈ℳ:d∣mN(d)=\#\m :d m\. Since d is odd, N(d)=⌊n/2d⌋N(d)= n/2d in the even case and N(d)=⌊n/d⌋−⌊n/2d⌋N(d)= n/d - n/2d in the odd case. In both cases N(d)=n2d+δd,|δd|≤1.N(d)= n2d+ _d, _d ≤ 1. For distinct p,q∈Pp,q∈ P we have p∣mq∣m=pq∣m1_\p m\1_\q m\=1_\pq m\, so expanding the square gives ∑m∈ℳ(ωP(m)−κ)2 _m ( _P(m)-κ )^2 =∑p,q∈Pp≠qN(pq)+(1−2κ)∑p∈PN(p)+κ2N(1) = _ subarraycp,q∈ P\\ p≠ q subarrayN(pq)+(1-2κ) _p∈ PN(p)+κ^2N(1) =n2(κ2−∑p∈P1p2)+nκ2−nκ2+nκ22+ℰ = n2 (κ^2- _p∈ P 1p^2 )+ nκ2-nκ^2+ nκ^22+E =nκ2−n2∑p∈P1p2+ℰ, = nκ2- n2 _p∈ P 1p^2+E, where ℰ=∑p,q∈Pp≠qδpq+(1−2κ)∑p∈Pδp+κ2δ1.E= _ subarraycp,q∈ P\\ p≠ q subarray _pq+(1-2κ) _p∈ P _p+κ^2 _1. Every p∈Pp∈ P is odd, so κ≤|P|/3κ≤ P /3 and therefore |ℰ|≤(|P|2−|P|)+(1+2κ)|P|+κ2≤(1+23+19)|P|2≤2|P|2. ≤ ( P ^2- P )+(1+2κ) P +κ^2≤ (1+ 23+ 19 ) P ^2≤ 2 P ^2. Discarding the negative term −n2∑p∈Pp−2- n2 _p∈ Pp^-2 completes the proof. ∎ C.4.3 Proof of the main theorem Proof of Theorem C.4.1. The lower bound is the set of odd numbers in [n][n], as noted above. For the upper bound, fix t≥1t≥ 1 and set z=n1/3,P=p an odd prime:t<p≤z,κ=∑p∈P1p.z=n^1/3, P=\p an odd prime:t<p≤ z\, κ= _p∈ P 1p. By Mertens’ theorem ∑p≤z1/p=loglogz+O(1) _p≤ z1/p= z+O(1), and loglogz=loglogn−log3 z= n- 3, so κ=loglogn+Ot(1)κ= n+O_t(1). Choose n0(t)n_0(t) so that κ≥max(1,12loglogn)κ≥ (1, 12 n ), n1/3≥4n^1/3≥ 4, and loglogn≤n n≤ n for all n≥n0(t)n≥ n_0(t). Let A⊆[n]A [n] be t-admissible and put B=[n]∖AB=[n] A. We claim that for each p∈Pp∈ P, #a∈A:a even,p∣a≤#b∈B:b odd,p∣b.\#\a∈ A:a even,\ p a\\ ≤\ \#\b∈ B:b odd,\ p b\. Indeed, let a∈Aa∈ A be even with p|ap a. As p is odd, a=2rpa=2rp for some r≥1r≥ 1, and a−p=(2r−1)pa-p=(2r-1)p is an odd multiple of p lying in [1,n][1,n]. The pair a−p<a-p<a has difference p>tp>t and p|ap a, so a−p∉Aa-p∉ A, that is, a−p∈Ba-p∈ B. The map a↦a−pa a-p is injective, which proves the claim. Summing the claim over p∈Pp∈ P gives ∑a∈Aa evenωP(a)≤∑b∈Bb oddωP(b). _ subarrayca∈ A\\ a even subarray _P(a)\ ≤\ _ subarraycb∈ B\\ b odd subarray _P(b). (13) Write x=#a∈A:a even,y=#b∈B:b odd.x=\#\a∈ A:a even\, y=\#\b∈ B:b odd\. Subtracting κ from every summand in equation 13 and rearranging, κ(x−y)≤∑b∈Bb odd(ωP(b)−κ)−∑a∈Aa even(ωP(a)−κ).κ(x-y)\ ≤\ _ subarraycb∈ B\\ b odd subarray ( _P(b)-κ )- _ subarrayca∈ A\\ a even subarray ( _P(a)-κ ). Apply Cauchy–Schwarz to each of the two sums, extend the resulting sums of squares to all integers of the relevant parity in [n][n], and invoke Lemma C.4.4: κ(x−y)≤(x+y)(nκ2+2|P|2)1/2.κ(x-y)\ ≤\ ( x+ y ) ( nκ2+2 P ^2 )^1/2. Now |P|≤z=n1/3 P ≤ z=n^1/3, and n1/3≥4≥4/κn^1/3≥ 4≥ 4/κ, so 2|P|2≤2n2/3≤nκ/22 P ^2≤ 2n^2/3≤ nκ/2. Moreover x≤n/2x≤ n/2 and y≤(n+1)/2y≤(n+1)/2, whence x+y≤2(n+1)≤3n x+ y≤ 2(n+1)≤ 3n. Therefore κ(x−y)≤3n⋅nκ(x-y)≤ 3n· nκ, that is, x−y≤n3κ.x-y≤ n 3κ. The set A has exactly ⌈n/2⌉−y n/2 -y odd elements and x even elements, so |A|=⌈n2⌉−y+x≤⌈n2⌉+n3κ≤n+12+n6loglogn≤n2+3nloglogn, A = n2 -y+x≤ n2 +n 3κ≤ n+12+n 6 n≤ n2+ 3n n, the last step because loglogn≤n n≤ n and 3−6>123- 6> 12. Since A was an arbitrary t-admissible subset of [n][n], this is the desired bound on F(n,t)F(n;t). ∎ References. Bloom (2026) T. F. Bloom. Erdős problem #635. https://w.erdosproblems.com/635, 2026. Accessed 12 August 2026. Elliott (2012) Peter DTA Elliott. Probabilistic number theory I: Mean-value theorems. Springer Science & Business Media, 2012. Guy (1983) Richard K. Guy. A miscellany of Erdős problems. The American Mathematical Monthly, 90(2):118–120, 1983. ISSN 00029890, 19300972. URL http://w.jstor.org/stable/2975810. Helfgott & Radziwił (2021) Harald Andres Helfgott and Maksym Radziwił. Expansion, divisibility and parity. arXiv preprint arXiv:2103.06853, 2021. Matomäki et al. (2016) Kaisa Matomäki, Maksym Radziwił, and Terence Tao. Sign patterns of the liouville and möbius functions. In Forum of Mathematics, Sigma, volume 4, p. e14. Cambridge University Press, 2016. Ruzsa (1999) Imre Z Ruzsa. Erdős and the integers. Journal of Number Theory, 79(1):115–163, 1999. C.5 Divisibility among binomial coefficients This problem asks for the density of the set of m for which some admissible k makes a product of k consecutive integers starting just above n divide the product of k consecutive integers starting just above m. Erdős & Straus 1977 asked whether all, or almost all, large m admit such a k, and if not what the density of those that do is; we show that the density exists and equals 11 for every fixed n≥2n≥ 2. The proof produces, for each B, a single k for which (n+kn) n+kn has no prime factor below B, so that the required divisibility is forced by one congruence condition on m modulo each prime divisor of that coefficient. C.5.1 Introduction For positive integers n and k write A(n,k)=(n+k)!/n!A(n,k)=(n+k)!/n! for the product of the k consecutive integers n+1,…,n+kn+1,…,n+k. Erdős & Straus 1977 study the divisibility relation A(n,k)|A(m,k)A(n,k) A(m,k) for m>nm>n. Since A(m,k)A(n,k)=(m+k)/(n+k), A(m,k)A(n,k)= m+kk / n+kk, that relation is exactly the divisibility (n+kn)|(m+k) n+kn\ |\ m+kk (14) between two binomial coefficients. Bounding k is essential here. Indeed, for k>m−nk>m-n one has A(m,k)A(n,k)=A(n+k,m−n)A(n,m−n), A(m,k)A(n,k)= A(n+k,m-n)A(n,m-n), and Erdős & Straus 1977 note that this quotient is an integer for k=A(n,m−n)−mk=A(n,m-n)-m, a value exceeding m−nm-n as soon as m≥n+2m≥ n+2. For m=n+1m=n+1 that value is 00; there one may take k=n+1k=n+1 instead, with quotient 22. Without a restriction on k the problem is therefore vacuous. With the natural restriction 1≤k≤m−n1≤ k≤ m-n in place, they ask the following question in their §1, in which the displayed divisibility is their (1.4). Given n>1n>1 is it true that for all (almost all) large m there exists a k, 1≤k≤m−n1≤ k≤ m-n so that (k+n)|(m+k) k+nn m+kk? If not, what is the density d∗(n)d^*(n) of integers m for which (1.4) has a solution with 1≤k≤m−n1≤ k≤ m-n? For a fixed integer n≥1n≥ 1 let Gn:=m∈ℕ:m>n,∃k∈ℕ, 1≤k≤m−n,(n+kn)|(m+k),G_n:= \m :\ m>n,\ ∃\,k ,\ 1≤ k≤ m-n,\ n+kn | m+kk \, and for a set S⊆ℕS write d(S):=limx→∞|S∩[1,x]|xd(S):= _x→∞ |S∩[1,x]|x for its natural density, when the limit exists, so that d∗(n)=d(Gn)d^*(n)=d(G_n). We prove that d∗(n)d^*(n) exists and equals 11 for every fixed n≥2n≥ 2, so that the “almost all” alternative holds. The case n=1n=1, excluded from the question above, is settled outright in Erdős & Straus 1977, and its proof is the carry argument used below. For a prime p, taking k=p−1k=p-1 turns equation 14 into the assertion p|(m+p−1p−1)p m+p-1p-1. The base-p digits of p−1p-1 are (p−1,0,0,…)(p-1,0,0,…), so adding m and p−1p-1 in base p produces a carry out of the units place as soon as p∤mp m, and Kummer’s theorem then gives p|(m+p−1p−1)p m+p-1p-1. No m>2m>2 is divisible by every prime p≤mp≤ m, and any prime p≤mp≤ m with p∤mp m supplies an admissible k=p−1k=p-1 in [1,m−1][1,m-1], so every m>2m>2 lies in G1G_1. Erdős and Straus write that already the next case, n=2n=2, “seems much more difficult to decide”. Erdős and Straus themselves prove two results about equation 14, both in the regime where m/nm/n is bounded. Erdős & Straus 1977 show that for fixed c>0c>0 and Λ>1 >1 only finitely many triples n,k,mn,k,m with k≥cnk≥ cn and n+k≤m≤Λn+k≤ m≤ n satisfy it, and their Theorem 3.3 removes the hypothesis k≥cnk≥ cn at the cost of requiring k≥2k≥ 2 and a prime in [n+1,n+k][n+1,n+k]. Both leave the range studied here untouched, since we fix n and let m→∞m→∞, so that m/n→∞m/n→∞. Apart from these, work on equation 14 has concentrated on the boundary case k=m−nk=m-n, that is, on the question whether for each fixed n some k satisfies (n+kn)|(n+2k) n+kn n+2kk. That case is raised in Erdős & Straus 1977, recorded by Erdös & Graham 1980, and appears as Problem 389 on the Erdős problems website (Bloom 2026), where it is listed as open; Ulas & Schinzel 2013 gives computational results for it and for a companion question of Erdős and Graham, verifying that a suitable k exists for every n≤20n≤ 20, and settling the companion Erdős–Graham question for every n≤9n≤ 9. Pomerance 2015 proves that for each fixed k≥1k≥ 1 the set of M with M+k|(2M)M+k 2MM has asymptotic density 11, and notes that the same proof yields density one for the divisibility (M+1)(M+2)⋯(M+k)∣(2M)(M+1)(M+2)·s(M+k) 2MM. Pomerance 2026 allows k to grow with M: the product (M+1)⋯(M+k)(M+1)·s(M+k) divides (2M) 2MM for all k≤ηlogMk≤η M, for any fixed η<1/log4η<1/ 4, and (M+k) M+kk divides (2M) 2MM for all k≤e0.8logMk≤ e^0.8 M, in both cases for a set of M of density 11. The mechanism there is the one we use: a high power of p dividing M+kM+k forces the low base-p digits of M to be large, and Kummer’s theorem converts this into carries. None of these statements implies our theorem: in each of them the dividend is the central binomial coefficient (2M) 2MM and k is small compared with M, whereas in equation 14 the dividend is (m+k) m+kk and the divisor is (n+kn) n+kn with n fixed and k unbounded. In a similar vein Harborth 1979 showed that for each fixed k almost all entries (Mj) Mj of Pascal’s triangle are divisible by M(M−1)⋯(M−k+1)M(M-1)·s(M-k+1); see the discussion in Pomerance 2015. We have found no prior source asserting that GnG_n has positive lower density, let alone density one. Throughout, p and q denote primes, vp(⋅)v_p(·) is the p-adic valuation, π(B)=∑p≤B1π(B)= _p≤ B1 is the prime-counting function, ϑ(B)=∑p≤Blogp (B)= _p≤ B p is Chebyshev’s function, and ω(N)ω(N) is the number of distinct prime divisors of N. Theorem C.5.1. For every fixed integer n≥2n≥ 2 the natural density d∗(n)d^*(n) exists and d∗(n)=1.d^*(n)=1. Since Gn⊆ℕG_n , its upper density is at most 11, so the entire content of the theorem is a lower bound on the lower density. Remark C.5.2. Whether all large m lie in GnG_n remains open, since we show that ℕ∖GnN G_n has density zero and not that it is finite. The proof is also silent about the least admissible k for a given m. To achieve density 1−ε1- it uses a single k=kBk=k_B of size eOn(B)e^O_n(B) and covers only those m with m≥n+kBm≥ n+k_B, so it gives no information about small k, and in particular none about the boundary case k=m−nk=m-n, which is Problem 389. We now summarize the proof. For each integer parameter B we exhibit one value k=kBk=k_B of the free variable in equation 14 that works simultaneously for every m in an explicit periodic set TBT_B, and we show that the density of TBT_B tends to 11 as B→∞B→∞. The key point is to choose kBk_B so that NB=(n+kBn)N_B= n+k_Bn has no prime factor below B. Every prime q|NBq N_B then divides exactly one of the n integers kB+1,…,kB+nk_B+1,…,k_B+n, say kB+i(q)k_B+i(q), and does so to the full multiplicity vq(NB)v_q(N_B). The single congruence condition mmodq≥i(q)m q≥ i(q) is then enough to force qvq(NB)|(m+kBkB)q^v_q(N_B) m+k_Bk_B, because kBk_B has residue qs−i(q)q^s-i(q) modulo qsq^s for every s≤vq(NB)s≤ v_q(N_B) and each such s therefore contributes a carry. Only Kummer’s theorem, Legendre’s formula, the Chinese remainder theorem and Chebyshev’s elementary bounds on π and ϑ are needed. C.5.2 A binomial coefficient free of small primes Fix n≥2n≥ 2. For an integer parameter B>nB>n define kB:=∏p≤Bpep,ep:=mine≥1:pe>n,k_B:= _p≤ Bp^e_p, e_p:= \e≥ 1:\ p^e>n\, (15) and set NB:=(n+kBn).N_B:= n+k_Bn. Lemma C.5.3. Every prime divisor of NBN_B is larger than B. Proof. Let p≤Bp≤ B. By equation 15 we have pep|kBp^e_p k_B, so the epe_p lowest base-p digits of kBk_B vanish, while n<pepn<p^e_p, so every base-p digit of n of index epe_p or higher vanishes. Adding n and kBk_B in base p therefore produces no carry at all: in each of the positions 0,…,ep−10,…,e_p-1 the digit of kBk_B is 00, and in each higher position the digit of n is 00, so no position ever sums to p or more. By Kummer’s theorem vp(n+kBn)v_p n+k_Bn equals the number of carries in this addition, whence vp(NB)=0.v_p(N_B)=0. Thus no prime p≤Bp≤ B divides NBN_B. ∎ Lemma C.5.4. Let q be a prime with q|NBq N_B and put aq:=vq(NB)a_q:=v_q(N_B). Then there is a unique index i(q)∈1,…,ni(q)∈\1,…,n\ with q|kB+i(q)q k_B+i(q), and for this index qaq|kB+i(q),q^a_q\ \|\ k_B+i(q), that is, aq=vq(kB+i(q))a_q=v_q (k_B+i(q) ). Proof. By Lemma C.5.3 we have q>B>nq>B>n, so q∤n!q n! and hence vq(NB)=vq(∏i=1n(kB+i))−vq(n!)=∑i=1nvq(kB+i).v_q(N_B)=v_q ( _i=1^n(k_B+i) )-v_q(n!)= _i=1^nv_q(k_B+i). The left-hand side is positive, so at least one summand is positive. Two distinct indices 1≤i<j≤n1≤ i<j≤ n with q|kB+iq k_B+i and q|kB+jq k_B+j would give q|j−iq j-i with 0<j−i<n<q0<j-i<n<q, which is impossible. Hence exactly one index i(q)i(q) contributes, and the displayed identity reduces to aq=vq(kB+i(q))a_q=v_q(k_B+i(q)). ∎ C.5.3 A congruence forcing the divisibility Lemma C.5.5. Let q|NBq N_B, write i=i(q)i=i(q) and a=aqa=a_q. If m is a nonnegative integer with mmodq∈i,i+1,…,q−1,m q∈\i,i+1,…,q-1\, then vq(m+kBkB)≥a.v_q m+k_Bk_B\ ≥\ a. Proof. Fix s with 1≤s≤a1≤ s≤ a. By Lemma C.5.4 we have qa|kB+iq^a k_B+i, hence qs|kB+iq^s k_B+i, so kB≡−i(modqs)k_B≡-i q^s. Since 1≤i≤n<q≤qs1≤ i≤ n<q≤ q^s, the residue of kBk_B modulo qsq^s is exactly qs−iq^s-i, that is, kB=usqs+(qs−i)for some integer us=⌊kB/qs⌋≥0.k_B=u_sq^s+(q^s-i) some integer u_s= k_B/q^s ≥ 0. Write m=vsqs+tsm=v_sq^s+t_s with 0≤ts<qs0≤ t_s<q^s. Since ts≡m(modq)t_s≡ m q and ts≥0t_s≥ 0, we have ts≥tsmodq=mmodq≥it_s≥ t_s q=m q≥ i. Consequently i≤ts<qs,soqs≤ts+qs−i< 2qs,i≤ t_s<q^s, q^s\ ≤\ t_s+q^s-i\ <\ 2q^s, and therefore ⌊m+kBqs⌋−⌊mqs⌋−⌊kBqs⌋=(vs+us)+⌊ts+qs−iqs⌋−vs−us=⌊ts+qs−iqs⌋=1. m+k_Bq^s - mq^s - k_Bq^s =(v_s+u_s)+ t_s+q^s-iq^s -v_s-u_s= t_s+q^s-iq^s =1. By Legendre’s formula applied to (m+kB)!(m+k_B)!, m!m! and kB!k_B!, vq(m+kBkB)=∑s≥1(⌊m+kBqs⌋−⌊mqs⌋−⌊kBqs⌋).v_q m+k_Bk_B= _s≥ 1 ( m+k_Bq^s - mq^s - k_Bq^s ). Each summand is nonnegative, since ⌊α+β⌋≥⌊α⌋+⌊β⌋ α+β ≥ α + β , and the a summands with 1≤s≤a1≤ s≤ a each equal 11. Hence the sum is at least a. ∎ Observe that the hypothesis of Lemma C.5.5 is a condition modulo q alone, even though its conclusion concerns the prime power qaq^a. This is what makes the Chinese remainder theorem cheap to apply. Define TB:=m∈ℕ:mmodq∈i(q),i(q)+1,…,q−1 for every prime q∣NB.T_B:= \m :\ m q∈\i(q),i(q)+1,…,q-1\\ for every prime q N_B \. (16) Corollary C.5.6. For every m∈TBm∈ T_B we have NB|(m+kBkB)N_B m+k_Bk_B. Consequently TB∩[n+kB,∞)⊆Gn.T_B∩[\,n+k_B,∞)\ \ G_n. Proof. Let m∈TBm∈ T_B. For each prime q|NBq N_B, Lemma C.5.5 gives vq(m+kBkB)≥aq=vq(NB)v_q m+k_Bk_B≥ a_q=v_q(N_B); hence NB|(m+kBkB)N_B m+k_Bk_B. If moreover m≥n+kBm≥ n+k_B, then k:=kBk:=k_B satisfies 1≤k≤m−n1≤ k≤ m-n and (n+kn)=NB n+kn=N_B divides (m+k) m+kk, so m∈Gnm∈ G_n. ∎ Example C.5.7. Take n=2n=2 and B=5B=5. Then e2=2e_2=2 and e3=e5=1e_3=e_5=1, so k5=4⋅3⋅5=60k_5=4· 3· 5=60 and N5=(622)=1891=31⋅61,N_5= 622=1891=31· 61, whose prime factors indeed exceed 55. Here 31|62=k5+231 62=k_5+2 and 61|61=k5+161 61=k_5+1, so i(31)=2i(31)=2 and i(61)=1i(61)=1, both with aq=1a_q=1. Corollary C.5.6 therefore says that (622)|(m+6060)whenever m≥62,m≢0,1(mod31),m≢0(mod61), 622\ |\ m+6060 m≥ 62, m ≡ 0,1 31, m ≡ 0 61, a set of density 2931⋅6061=17401891>0.92 2931· 6061= 17401891>0.92. The congruence conditions are sufficient but not necessary: for instance m=930m=930 is excluded by the condition at 3131, yet (622)|(99060) 622 99060. Indeed 930930 and 6060 have base-3131 digits (0,30)(0,30) and (29,1)(29,1), written from least to most significant, so their addition still carries out of the second place and 31|(99060)31 99060, while 930mod61=15≥i(61)930 61=15≥ i(61) leaves 930930 inside the condition at 6161. C.5.4 The density of the congruence set The definition equation 16 imposes congruence conditions modulo the finitely many distinct primes q|NBq N_B, so TBT_B is a union of residue classes modulo ∏q|NBq _q N_Bq and its natural density d(TB)d(T_B) exists. Since i(q)≤n<qi(q)≤ n<q, the condition at q excludes exactly the i(q)i(q) residues 0,1,…,i(q)−10,1,…,i(q)-1, so by the Chinese remainder theorem d(TB)=∏q|NB(1−i(q)q)≥∏q|NB(1−nq)> 0.d(T_B)= _q N_B (1- i(q)q )\ ≥\ _q N_B (1- nq )\ >\ 0. (17) Lemma C.5.8. For every integer B>nB>n we have logkB=On(B) k_B=O_n(B) and ω(NB)=On(BlogB).ω(N_B)=O_n\! ( B B ). Proof. By the minimality of epe_p in equation 15 we have pep−1≤np^e_p-1≤ n, whence eplogp≤logn+logpe_p p≤ n+ p, and therefore logkB=∑p≤Beplogp≤π(B)logn+ϑ(B). k_B= _p≤ Be_p p≤π(B) n+ (B). Chebyshev’s bounds π(B)=O(B/logB)π(B)=O(B/ B) and ϑ(B)=O(B) (B)=O(B) give logkB=On(B) k_B=O_n(B). For the second assertion, note that NB⋅n!=∏i=1n(kB+i)N_B· n!= _i=1^n(k_B+i), so every prime divisor of NBN_B divides some kB+ik_B+i with 1≤i≤n1≤ i≤ n. Fix such an i and let t be the number of distinct primes exceeding B that divide kB+ik_B+i. Each such prime is at least B+1B+1 because B is an integer, so their product divides kB+ik_B+i and is at least (B+1)t(B+1)^t, whence t≤log(kB+n)log(B+1).t≤ (k_B+n) (B+1). By Lemma C.5.3 every prime divisor of NBN_B exceeds B, so summing over the n values of i gives ω(NB)≤nlog(kB+n)log(B+1)=On(BlogB),ω(N_B)≤ n\, (k_B+n) (B+1)=O_n\! ( B B ), where the last step uses logkB=On(B) k_B=O_n(B). ∎ Lemma C.5.9. We have d(TB)→1d(T_B)→ 1 as B→∞B→∞. Proof. Assume that B≥2nB≥ 2n. Every prime q|NBq N_B satisfies q≥B+1>2nq≥ B+1>2n by Lemma C.5.3 and the integrality of B, so n/q<1/2n/q<1/2. For 0≤x≤1/20≤ x≤ 1/2 one has log(1−x)≥−2x (1-x)≥-2x; indeed h(x):=log(1−x)+2xh(x):= (1-x)+2x satisfies h(0)=0h(0)=0 and h′(x)=2−11−x≥0h (x)=2- 11-x≥ 0 on [0,1/2][0,1/2], so h≥0h≥ 0 there. Applying this to each factor of equation 17, logd(TB)≥∑q|NBlog(1−nq)≥−2∑q|NBnq≥−2nω(NB)B+1, d(T_B)\ ≥\ _q N_B (1- nq )\ ≥\ -2 _q N_B nq\ ≥\ - 2n\,ω(N_B)B+1, where the last step uses q≥B+1q≥ B+1 for every such q. By Lemma C.5.8 the right-hand side is On(1/logB)O_n(1/ B) in absolute value, hence logd(TB)→0 d(T_B)→ 0 and d(TB)→1d(T_B)→ 1. ∎ Note that d(TB)d(T_B) need not increase with B. Continuing Example C.5.7, we have k13=4⋅3⋅5⋅7⋅11⋅13=60060k_13=4· 3· 5· 7· 11· 13=60060 and N13=(600622)=17⋅59⋅509⋅3533,N_13= 600622=17· 59· 509· 3533, with i(17)=i(3533)=1i(17)=i(3533)=1 and i(59)=i(509)=2i(59)=i(509)=2, so that equation 17 gives d(T13)=1617⋅5759⋅507509⋅35323533=0.9054…< 0.9201…=d(T5).d(T_13)= 1617· 5759· 507509· 35323533=0.9054…\ <\ 0.9201…=d(T_5). Raising B from 55 to 1313 has replaced two prime factors by four, one of them barely above B. Only the limit is asserted. C.5.5 Proof of the main theorem Proof of Theorem C.5.1. Let ε>0 >0. By Lemma C.5.9 choose an integer B≥2nB≥ 2n with d(TB)>1−εd(T_B)>1- . By Corollary C.5.6 every element of TBT_B that is at least n+kBn+k_B lies in GnG_n, so for every x≥n+kBx≥ n+k_B, |Gn∩[1,x]|≥|TB∩[n+kB,x]|≥|TB∩[1,x]|−(n+kB).|G_n∩[1,x]|\ ≥\ |T_B∩[n+k_B,x] |\ ≥\ |T_B∩[1,x]|-(n+k_B). The subtracted quantity is independent of x, so dividing by x and letting x→∞x→∞ gives lim infx→∞|Gn∩[1,x]|x≥limx→∞|TB∩[1,x]|x=d(TB)>1−ε. _x→∞ |G_n∩[1,x]|x\ ≥\ _x→∞ |T_B∩[1,x]|x=d(T_B)>1- . Since ε>0 >0 was arbitrary, the lower density of GnG_n is 11, while its upper density is trivially at most 11. Hence the natural density exists and d∗(n)=1d^*(n)=1, completing the proof. ∎ References. Bloom (2026) T. F. Bloom. Erdős problem #389. https://w.erdosproblems.com/389, 2026. Accessed 12 August 2026. Erdős & Straus (1977) P. Erdős and E. G. Straus. On products of consecutive integers. In Number Theory and Algebra, p. 63–70. Academic Press, New York, 1977. Erdös & Graham (1980) Paul Erdös and Ronald L Graham. Old and new problems and results in combinatorial number theory, volume 28. L’Enseignement Mathematiques Un. Geneve, 1980. Harborth (1979) Heiko Harborth. Divisibility of. The American Mathematical Monthly, 86(2):115–117, 1979. Pomerance (2015) Carl Pomerance. Divisors of the middle binomial coefficient. The American Mathematical Monthly, 122(7):636–644, 2015. Pomerance (2026) Carl Pomerance. Remarks on the middle binomial coefficient. Integers, 26, 2026. Ulas & Schinzel (2013) Maciej Ulas and Andrzej Schinzel. A note on erdős–straus and erdős–graham divisibility problems (with an appendix by andrzej schinzel). International Journal of Number Theory, 9(03):583–599, 2013. C.6 Many-one GapP-completeness for binary symmetric group characters This problem concerns the complexity of evaluating an irreducible character of the symmetric group when both partitions are written in binary. Ikenmeyer et al. 2024 proved that this evaluation is GapP-complete under Turing reductions and conjectured that many-one reductions already suffice; we prove the conjecture, in the sharper form that the partition indexing the character may always be taken to have at most two parts. The reduction encodes an arbitrary GapP function as a subset-sum count in a large base and reads the value off a two-row character, which by a classical identity is a difference of two such counts at consecutive targets. C.6.1 Introduction For a partition λ⊢nλ n let χλχ^λ denote the irreducible character of the symmetric group n S_n indexed by λ. Its value at a permutation depends only on the cycle type of that permutation, so for μ⊢nμ n we write χλ(μ)χ^λ(μ) for the common value of χλχ^λ on the conjugacy class of cycle type μ. The problem ComputeCharBinary takes as input two partitions λ,μ⊢nλ,μ n, each presented as a list of parts written in binary, and outputs the integer χλ(μ)χ^λ(μ). Because a character value can be negative, the natural home for this function is not the counting class #\# P of Valiant 1979 but the gap class =#−# GapP=\# P-\# P of differences of two #\# P functions, introduced by Fenner et al. 1994. A function F is GapP-hard under many-one reductions if for every f∈f∈ GapP there is a polynomial-time computable map R with f(x)=F(R(x))f(x)=F(R(x)) for all x, and GapP-complete if moreover F∈F∈ GapP. This is stronger than hardness under Turing reductions, where f is only required to be computable in polynomial time given an oracle for F; we follow Papadimitriou 2003 for the standard conventions. Ikenmeyer et al. 2024 proved that deciding χλ(μ)=0χ^λ(μ)=0 is = C_= P-complete and that deciding χλ(μ)≥0χ^λ(μ)≥ 0 is P-complete, both under many-one reductions, and deduced that neither |χλ(μ)||χ^λ(μ)| nor χλ(μ)2χ^λ(μ)^2 lies in #\# P unless the polynomial hierarchy collapses to its second level. As a byproduct of the same reduction they obtained that ComputeCharBinary is GapP-complete under Turing reductions (Ikenmeyer et al. 2024), and they noted that their route cannot be pushed as far as a parsimonious reduction. They then stated the following (Ikenmeyer et al. 2024, Conjecture 5.2). The problem ComputeCharBinary is GapP-complete under many-one reductions. We prove this conjecture, in the sharper form that the reduction may always be taken to output a partition λ with at most two parts. The closest previous result is the Turing-reduction completeness stated above. Before that, Hepler 1994 proved that computing χλ(μ)χ^λ(μ) is #\# P-hard under many-one reductions already for unary input, hence also for binary input; #\# P-hardness constrains only the nonnegative part of the character and does not by itself give GapP-hardness. Pak & Panova 2017 showed that the positivity of a Kronecker coefficient can be decided in time O(logN)O( N) for partitions with a bounded number of parts and largest part N, while Ikenmeyer et al. 2017 showed that deciding positivity of a Kronecker coefficient is NP-hard; Panova 2023 surveys this circle of questions. On the algorithmic side, Bravyi et al. 2025 give an algorithm computing a matrix product state that encodes the column (χλ(μ))λ⊢n(χ^λ(μ))_λ n of the character table, and record the worst-case #\# P-hardness of a single entry as the obstruction to a general polynomial-time algorithm. We are not aware of a previous many-one GapP-hardness result for this function. Theorem C.6.1. The function ComputeCharBinary is GapP-complete under polynomial-time many-one reductions. More precisely, for every f∈f∈ GapP there is a polynomial-time computable map x↦(λx,μx)x ( _x, _x), whose values are pairs of partitions of a common integer nxn_x with λx _x having at most two parts, such that f(x)=χλx(μx)f(x)=χ _x( _x) for every input x. Remark C.6.2. The reduction below computes f(x)f(x) exactly, in the many-one sense of the definition above, but it exhibits the value as a difference of two subset-sum counts rather than as a single unsigned count, so it gives no combinatorial interpretation of the character value. Ikenmeyer et al. 2024 record a different obstruction to parsimoniousness in their own reduction, namely that its passage from matchings to counts of ordered set partitions multiplies the count by a fixed factor, since each matching arises from the same number of ordered set partitions, obtained from one another by permuting the bins and the repeated items. The binary encoding is also essential. For unary input the quantity Nμ(t)N_μ(t) defined below obeys a subset-sum recursion over the parts of μ with 0≤t≤n0≤ t≤ n, so two-row character values are then computable in polynomial time, and the family of instances produced below can carry no hardness at all in the unary model unless every GapP function is polynomial-time computable. An alternative reduction runs through the identities of Ikenmeyer et al. 2024, which realize as a single character value the difference of two counts of ordered set partitions with the same item sizes and with bin sizes agreeing outside two distinguished bins, whose sizes are 22 and 44 in the first count and 11 and 55 in the second; the two-row route above avoids the padding that this requires. The proof rests on one classical identity and one gadget. For a two-row shape the character value is a difference of two subset-sum counts at consecutive targets, χ(n−s,s)(μ)=Nμ(s)−Nμ(s−1),χ^(n-s,s)(μ)=N_μ(s)-N_μ(s-1), where Nμ(t)N_μ(t) is the number of subsets of the parts of μ with total size t. It therefore suffices to manufacture, out of two given counting problems, a single multiset of binary integers whose subset sums realize the first count at s and the second at s−1s-1. The key point is that the two targets differ by exactly 11, so the two problems have to be separated inside a single units digit. We take the two problems to be counts of exact covers of a finite set, write every part in a large base Q, give the two instances disjoint blocks of base-Q digits, and adjoin two selector parts that differ in the units digit and each of which pre-fills the digit block of the other instance. C.6.2 Characters of two-row shape Throughout, a partition μ=(μ1,…,μm)⊢nμ=( _1,…, _m) n has positive parts, and parts of equal size are regarded as distinct, indexed by [m][m]. For t∈ℤt set Nμ(t):=#I⊆[m]:∑i∈Iμi=t,N_μ(t):=\# \I [m]: _i∈ I _i=t \, so that Nμ(t)=0N_μ(t)=0 for t<0t<0 and Nμ(0)=1N_μ(0)=1. Lemma C.6.3. Let μ⊢nμ n and let 0≤s≤n/20≤ s≤ n/2. Then χ(n−s,s)(μ)=Nμ(s)−Nμ(s−1).χ^(n-s,s)(μ)=N_μ(s)-N_μ(s-1). Proof. For an integer vector α=(α1,α2,…)α=( _1, _2,…) with entries summing to n let ϕαφ^α be the character of the representation of n S_n induced from the trivial representation of the Young subgroup α1×α2×⋯ S_ _1× S_ _2×·s, with the convention ϕα=0φ^α=0 when some αi<0 _i<0. Equivalently, ϕαφ^α is the character of the action of n S_n on the words containing exactly αi _i letters equal to i. Such a word is fixed by a permutation π precisely when it is constant on each cycle of π, so for π of cycle type μ the value ϕα(μ)φ^α(μ) is the number of ways to label the m cycles by letters so that the cycles labeled i have total length αi _i (Ikenmeyer et al. 2024). In particular, for a vector with two entries, ϕ(n−t,t)(μ)=Nμ(t).φ^(n-t,t)(μ)=N_μ(t). The Frobenius character formula (James 2006, Eq. 2.3.8), equivalently Young’s rule (Sagan 2001), gives χ(n−s,s)=ϕ(n−s,s)−ϕ(n−s+1,s−1),χ^(n-s,s)=φ^(n-s,s)-φ^(n-s+1,s-1), and evaluating at cycle type μ yields the identity. ∎ C.6.3 A difference of two exact cover counts An instance of #ExactCover\# ExactCover is a pair (X,C)(X,C) in which X=[k]X=[k] with k≥0k≥ 0 and C=(S1,…,Sr)C=(S_1,…,S_r) is a list of nonempty subsets of X, members of equal content being regarded as distinct and indexed by [r][r]. A selection is a subset F⊆[r]F [r], and it is an exact cover if every u∈Xu∈ X lies in SiS_i for exactly one i∈Fi∈ F. The value of the instance is the number ec(X,C):=#F⊆[r]:F is an exact coverec(X,C):=\#\F [r]:F is an exact cover\ of exact covers, so that ec(∅,())=1ec( ,())=1, the empty selection covering the empty ground set. Deciding whether an exact cover exists is NP-complete already when every member of C has three elements (Garey & Johnson 2002); we put no bound on the sizes of the members, which the gadget below does not need. An element of X lying in no member of C is covered by no selection, so an instance containing such an element has value 00. Replacing every such instance by the fixed instance ([3],(1,2,2,3)) ([3],(\1,2\,\2,3\) ), whose value is also 00, we may and do assume k≤∑S∈C|S|,k≤ _S∈ C|S|, so that k is bounded by the length of the instance. Lemma C.6.4. There is a polynomial-time computable map sending a conjunctive normal form formula φ with clauses of between one and three literals to an instance (Xφ,Cφ)(X_ ,C_ ) of #ExactCover\# ExactCover with ec(Xφ,Cφ)ec(X_ ,C_ ) equal to the number of satisfying assignments of φ . Proof. Let φ have variables x1,…,xNx_1,…,x_N and clauses c1,…,cMc_1,…,c_M, and let cjc_j be the disjunction of the literals ℓj,1,…,ℓj,wj _j,1,…, _j,w_j with 1≤wj≤31≤ w_j≤ 3; repeated and complementary literals inside a clause are allowed. Take the ground set Xφ:=vi:i∈[N]∪zj:j∈[M]∪pj,q:j∈[M],q∈[wj],X_ :=\v_i:i∈[N]\∪\z_j:j∈[M]\∪\p_j,q:j∈[M],\ q∈[w_j]\, one element for each variable, one for each clause and one for each position inside a clause, and let CφC_ consist of the sets Vi,t:=vi∪pj,q:ℓj,q∈xi,¬xi is falsified by xi=t(i∈[N],t∈0,1)V_i,t:=\v_i\∪\p_j,q: _j,q∈\x_i, x_i\ is falsified by x_i=t\ (i∈[N],\ t∈\0,1\) together with the sets Wj,T:=zj∪pj,q:q∈T(j∈[M],∅≠T⊆[wj]).W_j,T:=\z_j\∪\p_j,q:q∈ T\ (j∈[M],\ ≠ T [w_j]). All of them are nonempty, there are 2N+∑j(2wj−1)≤2N+7M2N+ _j(2^w_j-1)≤ 2N+7M of them, and the list is written down in time linear in the length of φ . Let F be an exact cover. The element viv_i lies only in Vi,0V_i,0 and Vi,1V_i,1, so exactly one of the two is selected; let α(i)α(i) be the corresponding value of xix_i. The element zjz_j lies only in the sets Wj,TW_j,T, so for each j exactly one T=TjT=T_j is selected. The selected sets Vi,α(i)V_i,α(i) cover between them precisely those pj,qp_j,q whose literal is false under α, so exactness at the elements pj,qp_j,q forces TjT_j to be the set of positions q with ℓj,q _j,q true under α. As TjT_j is nonempty, α satisfies every clause. Conversely, let α satisfy φ and let TjT_j be the set of positions of cjc_j holding a literal true under α, which is nonempty. The sets Vi,α(i)V_i,α(i) with i∈[N]i∈[N] and Wj,TjW_j,T_j with j∈[M]j∈[M] then cover each element of XφX_ exactly once. The two constructions are mutually inverse, so the exact covers correspond bijectively to the satisfying assignments of φ . ∎ Given two instances of #ExactCover\# ExactCover, write Diff#ExactCover((X,C),(X′,C′)):=ec(X,C)−ec(X′,C′). Diff\# ExactCover ((X,C),(X ,C ) ):=ec(X,C)-ec(X ,C ). Lemma C.6.5. The function Diff#ExactCover Diff\# ExactCover is GapP-hard under polynomial-time many-one reductions. Proof. Let f∈f∈ GapP and write f=g−hf=g-h with g,h∈#g,h∈\# P. If g(x)=#w∈0,1q(|x|):V(x,w)=1g(x)=\#\w∈\0,1\^q(|x|):V(x,w)=1\ for a polynomial q and a polynomial-time predicate V, then the standard simulation of a polynomial-time machine by a Boolean circuit (Papadimitriou 2003) produces in polynomial time a circuit Γx _x with gates of fan-in at most two and with #CircuitSat(Γx)=g(x)\# CircuitSat( _x)=g(x); hence #CircuitSat\# CircuitSat is #\# P-complete under parsimonious reductions. The Tseitin transformation (Tseitin 1983) turns a circuit into a formula in conjunctive normal form with clauses of between one and three literals: it introduces one variable for each gate, adds the clauses forcing that variable to equal the value of the gate, and adds a unit clause forcing the output gate to take the value 11. The gate variables are determined by the input variables, so every satisfying assignment of the circuit extends to exactly one satisfying assignment of the formula, and the transformation is parsimonious. Composing it with Lemma C.6.4 and normalizing as above, we obtain polynomial-time computable instances with ec(Xx,Cx)=g(x),ec(Xx′,Cx′)=h(x),ec(X_x,C_x)=g(x), (X _x,C _x)=h(x), and therefore f(x)=Diff#ExactCover((Xx,Cx),(Xx′,Cx′))f(x)= Diff\# ExactCover((X_x,C_x),(X _x,C _x)). ∎ C.6.4 The digit gadget The construction is a subset-sum encoding in a large base, in the style of the strong NP-hardness proof of 44-Partition (Garey & Johnson 2002). Fix two instances (X,C)(X,C) and (X′,C′)(X ,C ) as above, with X=[k]X=[k], X′=[k′]X =[k ], C=(S1,…,Sr)C=(S_1,…,S_r) and C′=(S1′,…,Sr′)C =(S _1,…,S _r ), and set Q:=r+r′+4.Q:=r+r +4. We index base-Q digit positions by 0,1,…,k+k′+10,1,…,k+k +1, and call position 00 the units digit, positions 1,…,k1,…,k the X-block, positions k+1,…,k+k′k+1,…,k+k the X′X -block, and position k+k′+1k+k +1 the selector digit. For S∈CS∈ C and S′∈C′S ∈ C put aS:=∑u∈SQu,bS′:=∑u∈S′Qk+u,a_S:= _u∈ SQ^u, b_S := _u∈ S Q^k+u, so that aSa_S marks the elements covered by S inside the X-block, and likewise for bS′b_S . Let A:=∑u∈XQu,B:=∑u∈X′Qk+uA:= _u∈ XQ^u, B:= _u∈ X Q^k+u be the all-ones patterns on the two blocks, and let D:=Qk+k′+1D:=Q^k+k +1. Introduce the two selectors and the guard cA:=B+D+1,cB:=A+D,H:=A+B+D+2,c_A:=B+D+1, c_B:=A+D, H:=A+B+D+2, and set s:=A+B+D+1,s:=A+B+D+1, so that H=s+1H=s+1. Let μ be the partition obtained by sorting the multiset aS:S∈C∪bS′:S′∈C′∪cA,cB,H\a_S:S∈ C\∪\b_S :S ∈ C \∪\c_A,c_B,H\ into weakly decreasing order, let n:=|μ|n:=|μ|, and put λ:=(n−s,s).λ:=(n-s,s). Table 5 displays the base-Q digits of every quantity involved. units digit X-block X′X -block selector digit Q0Q^0 Q1,…,QkQ^1,…,Q^k Qk+1,…,Qk+k′Q^k+1,…,Q^k+k Qk+k′+1Q^k+k +1 aSa_S (S∈C)(S∈ C) 00 S1_S 0 00 bS′b_S (S′∈C′)(S ∈ C ) 00 0 S′1_S 00 cAc_A 11 0 1 11 cBc_B 00 1 0 11 H 22 1 1 11 s 11 1 1 11 s−1s-1 00 1 1 11 Table 5: The base-Q digits of the parts of μ and of the two targets s and s−1s-1. Here 1 and 0 denote the all-ones and all-zeros patterns on the block indicated, and S1_S denotes the indicator pattern of S on the X-block, with S′1_S defined analogously on the X′X -block. No column can reach Q, so subset sums may be compared digit by digit: a subset of parts summing to s must take cAc_A, hence miss the X′X -block entirely and pick out an exact cover of (X,C)(X,C), while a subset summing to s−1s-1 must take cBc_B and pick out an exact cover of (X′,C′)(X ,C ). Lemma C.6.6. Let m=r+r′+3m=r+r +3 be the number of parts of μ. For every I⊆[m]I [m] the base-Q digits of ∑i∈Iμi _i∈ I _i are obtained by adding the digit patterns of the parts indexed by I, without carrying. Consequently, for t∈s−1,st∈\s-1,s\ one has ∑i∈Iμi=t _i∈ I _i=t if and only if those digit patterns sum to the digit pattern of t. Proof. By Table 5, every part of μ has all its base-Q digits in 0,1,2\0,1,2\. At a position of the X-block the parts with a nonzero digit there are among the r parts aSa_S, the selector cBc_B and the guard H, each contributing 11, so the total over any I is at most r+2r+2. Symmetrically a position of the X′X -block receives at most r′+2r +2. The selector digit receives at most 33, from cAc_A, cBc_B and H, and the units digit receives at most 33, namely 11 from cAc_A and 22 from H. Every other position receives 00. All of these totals are smaller than Q=r+r′+4Q=r+r +4, so no carrying occurs. Finally s has digit 11 at the units digit, at every position of the two blocks, and at the selector digit, while s−1s-1 has the same digits except 00 at the units digit; both patterns have entries below Q, so two such integers agree if and only if their digit patterns agree. ∎ Lemma C.6.7. The pair (λ,μ)(λ,μ) is a valid input to ComputeCharBinary, with λ a partition of n having at most two parts and 0≤s≤n/20≤ s≤ n/2, and it is computable from (X,C)(X,C) and (X′,C′)(X ,C ) in polynomial time. Proof. Since cA+cB+H=2A+2B+3D+3c_A+c_B+H=2A+2B+3D+3, we get n=∑S∈CaS+∑S′∈C′bS′+2A+2B+3D+3,n= _S∈ Ca_S+ _S ∈ C b_S +2A+2B+3D+3, and therefore n−2s=∑S∈CaS+∑S′∈C′bS′+D+1>0.n-2s= _S∈ Ca_S+ _S ∈ C b_S +D+1>0. Hence n−s>s≥0n-s>s≥ 0, so λ=(n−s,s)λ=(n-s,s) is a partition of n with at most two parts and s≤n/2s≤ n/2. For the running time, the normalization gives k≤∑S∈C|S|k≤ _S∈ C|S| and k′≤∑S′∈C′|S′|k ≤ _S ∈ C |S |, so the largest exponent k+k′+1k+k +1 is linear in the length of the input, while log2Q=O(log(r+r′+4)) _2Q=O( (r+r +4)). Thus each of the r+r′+3r+r +3 parts of μ, and each of n and s, is an integer of O((k+k′+1)log(r+r′+4))O ((k+k +1) (r+r +4) ) bits, and all of them are produced by polynomially many arithmetic operations on integers of that size. ∎ Lemma C.6.8. For the partition μ constructed above, Nμ(s)=ec(X,C),Nμ(s−1)=ec(X′,C′).N_μ(s)=ec(X,C), N_μ(s-1)=ec(X ,C ). Proof. Let I index a subset of the parts with ∑i∈Iμi∈s−1,s _i∈ I _i∈\s-1,s\, which by Lemma C.6.6 we may analyze digit by digit. Since H=s+1>sH=s+1>s, the guard is not taken. Both s and s−1s-1 have selector digit 11, and apart from H only cAc_A and cBc_B have a nonzero selector digit, each equal to 11; hence exactly one of cAc_A and cBc_B is taken. Suppose first that ∑i∈Iμi=s _i∈ I _i=s. The units digit of s is 11, and among the remaining parts only cAc_A has a nonzero units digit, so cAc_A is taken and cBc_B is not. Now cAc_A already contributes 11 at every position of the X′X -block, which is exactly the digit of s there, so no part bS′b_S is taken. The parts taken are therefore cA∪aSi:i∈F\c_A\∪\a_S_i:i∈ F\ for some F⊆[r]F [r], and the one remaining requirement is that ∑i∈FaSi _i∈ Fa_S_i have digit 11 at every position of the X-block. By the definition of aSa_S this says exactly that the selected sets cover each element of the ground set exactly once, that is, that F is an exact cover of (X,C)(X,C). Hence the subsets of parts summing to s correspond bijectively to the exact covers of (X,C)(X,C), and Nμ(s)=ec(X,C)N_μ(s)=ec(X,C). Suppose now that ∑i∈Iμi=s−1 _i∈ I _i=s-1. The units digit of s−1s-1 is 00, so cAc_A is not taken, and therefore cBc_B is. Then cBc_B contributes 11 at every position of the X-block, so no part aSa_S is taken, and the parts taken are cB∪bSi′:i∈F′\c_B\∪\b_S _i:i∈ F \ with F′⊆[r′]F [r ] subject to the same condition on the X′X -block. Hence Nμ(s−1)=ec(X′,C′)N_μ(s-1)=ec(X ,C ). ∎ C.6.5 Membership in GapP Proposition C.6.9. ComputeCharBinary belongs to GapP. Proof. Let the input be λ=(λ1,…,λp)λ=( _1,…, _p) and μ=(μ1,…,μm)μ=( _1,…, _m), both partitions of n written as binary lists of parts, so that p, m and all the bit lengths involved are bounded by the length of the input; an input not of this form is recognized in polynomial time and given the value 00. For an integer vector β=(β1,…,βp)β=( _1,…, _p) let Pμ(β)P_μ(β) denote the number of ordered tuples (K1,…,Kp)(K_1,…,K_p) of pairwise disjoint, possibly empty subsets of [m][m] whose union is [m][m], such that ∑j∈Kiμj=βi _j∈ K_i _j= _i for every i; this is 00 if some βi<0 _i<0, because the parts of μ are positive. As in the proof of Lemma C.6.3 one has ϕβ(μ)=Pμ(β)φ^β(μ)=P_μ(β), so the Frobenius character formula (James 2006, Eq. 2.3.8) reads χλ(μ)=∑σ∈psgn(σ)Pμ(λ+σ−id),χ^λ(μ)= _σ∈ S_psgn(σ)\,P_μ(λ+σ-id), where λ+σ−idλ+σ-id is the vector whose iith entry is λi+σ(i)−i _i+σ(i)-i; see also Ikenmeyer et al. 2024. Let g and h count the pairs (σ,K)(σ,K) in which σ∈pσ∈ S_p has sgn(σ)=+1sgn(σ)=+1 and sgn(σ)=−1sgn(σ)=-1 respectively, and K=(K1,…,Kp)K=(K_1,…,K_p) is a tuple as above whose blocks have μ-weights ∑j∈Kiμj _j∈ K_i _j equal to the entries of λ+σ−idλ+σ-id. Such a pair is specified by O((p+m)log(p+1))O ((p+m) (p+1) ) bits and is verified by computing sgn(σ)sgn(σ) and comparing p sums of binary integers of O(logn)O( n) bits each, so g,h∈#g,h∈\# P. Therefore χλ(μ)=g−hχ^λ(μ)=g-h lies in GapP. ∎ C.6.6 Proof of the main theorem Proof of Theorem C.6.1. Membership in GapP is Proposition C.6.9. For hardness, fix f∈f∈ GapP. By Lemma C.6.5 there is a polynomial-time computable map sending an input x to two exact cover instances with f(x)=ec(Xx,Cx)−ec(Xx′,Cx′).f(x)=ec(X_x,C_x)-ec(X _x,C _x). Apply the digit gadget to those two instances, and let (λx,μx)( _x, _x) be the resulting pair, with λx=(nx−sx,sx) _x=(n_x-s_x,s_x). By Lemma C.6.7 the map x↦(λx,μx)x ( _x, _x) is computable in polynomial time, its values are partitions of nxn_x, the shape λx _x has at most two parts, and 0≤sx≤nx/20≤ s_x≤ n_x/2. By Lemma C.6.3 and Lemma C.6.8, χλx(μx)=Nμx(sx)−Nμx(sx−1)=ec(Xx,Cx)−ec(Xx′,Cx′)=f(x).χ _x( _x)=N_ _x(s_x)-N_ _x(s_x-1)=ec(X_x,C_x)-ec(X _x,C _x)=f(x). Hence every f∈f∈ GapP many-one reduces to ComputeCharBinary, completing the proof. ∎ References. Bravyi et al. (2025) Sergey Bravyi, David Gosset, Vojtech Havlicek, and Louis Schatzki. Classical and quantum algorithms for characters of the symmetric group. PRX Quantum, 6(3):030323, 2025. Fenner et al. (1994) Stephen A Fenner, Lance J Fortnow, and Stuart A Kurtz. Gap-definable counting classes. Journal of Computer and System Sciences, 48(1):116–148, 1994. Garey & Johnson (2002) Michael R Garey and David S Johnson. Computers and intractability, volume 29. wh freeman New York, 2002. Hepler (1994) Charles Thomas Hepler. On the complexity of computing characters of finite groups. Master’s thesis, University of Calgary, 1994. Ikenmeyer et al. (2017) Christian Ikenmeyer, Ketan D Mulmuley, and Michael Walter. On vanishing of kronecker coefficients. computational complexity, 26(4):949–992, 2017. Ikenmeyer et al. (2024) Christian Ikenmeyer, Igor Pak, and Greta Panova. Positivity of the symmetric group characters is as hard as the polynomial time hierarchy. International Mathematics Research Notices, 2024(10):8442–8458, 2024. James (2006) Gordon Douglas James. The representation theory of the symmetric groups. Springer, 2006. Pak & Panova (2017) Igor Pak and Greta Panova. On the complexity of computing kronecker coefficients. computational complexity, 26(1):1–36, 2017. Panova (2023) Greta Panova. Computational complexity in algebraic combinatorics. arXiv preprint arXiv:2306.17511, 2023. Papadimitriou (2003) Christos H Papadimitriou. Computational complexity. In Encyclopedia of Computer Science, p. 260–265. John Wiley and Sons, 2003. Sagan (2001) Bruce Sagan. The symmetric group: representations, combinatorial algorithms, and symmetric functions, volume 203. Springer Science & Business Media, 2001. Tseitin (1983) Grigori S Tseitin. On the complexity of derivation in propositional calculus. In Automation of reasoning: 2: Classical papers on computational logic 1967–1970, p. 466–483. Springer, 1983. Valiant (1979) Leslie G Valiant. The complexity of computing the permanent. Theoretical computer science, 8(2):189–201, 1979. C.7 Matching variance versus the residual matching number Kahn’s normal law for matchings characterizes asymptotic normality of the size of a uniformly random matching through five graph statistics that are bounded or unbounded together, and asks how tightly two of them, the variance σ2σ^2 and the residual matching number λ, are tied to one another. We show that in one direction they are not tied at all: along cliques carrying private pendant leaves the ratio σ2/λσ^2/λ grows at least linearly in the number of leaves at each clique vertex, while both parameters tend to infinity. The mechanism is that extra leaves inflate the fluctuation of the clique matching without inflating the residue. C.7.1 Introduction For a finite simple graph G let ℳ(G)M(G) be the set of matchings of G, let M be drawn uniformly from ℳ(G)M(G), and put ξG=|M| _G=|M|. Write μ(G)=[ξG]μ(G)=E[ _G] and σ2(G)=Var[ξG]σ^2(G)=Var[ _G], and write ν(G)ν(G) and τ(G)τ(G) for the matching number and the vertex cover number of G. For x∈V(G)x∈ V(G) let p(x)p(x) denote the probability that x is not covered by M. By a theorem of Godsil 1981, resting on the real-rootedness of the matching polynomial of Heilmann & Lieb 1972, the distribution of ξGn _G_n along a sequence of graphs is asymptotically normal if and only if σ(Gn)→∞σ(G_n)→∞. Kahn 2000 identified four combinatorial statistics with exactly the same threshold behavior, two of which are introduced there for the purpose. The first is the cover defect κ(G)=min∑y∈Yp(y):Y a vertex cover of G,κ(G)= \ _y∈ Yp(y):\ Y a vertex cover of G \, and the second, writing FMF_M for the set of edges of G meeting no edge of M, is the residual matching number λ(G)=[ν(FM)].λ(G)=E [ν(F_M) ]. Thus FMF_M is the edge set induced by the vertices left uncovered by M, and λ measures how much of a matching still fits into the residue. Kahn’s theorem states that for any sequence (Gn)(G_n) with |V(Gn)|→∞|V(G_n)|→∞ and δ(Gn)≥1δ(G_n)≥ 1, and with σn=σ(Gn) _n=σ(G_n) and so on, the five conditions σn=O(1),νn−μn=O(1),τn−μn=O(1),κn=O(1),λn=O(1) _n=O(1), _n- _n=O(1), _n- _n=O(1), _n=O(1), _n=O(1) are equivalent (Kahn 2000, Theorem 1.10). He then asks how tight the comparison between the first and the last of these is (Kahn 2000, Question 7.3): How closely related are σ2σ^2 and λ? In particular, is it true that λ=Θ(σ2)λ= (σ^2) (that is, are there bounds on the ratios λ/σ2λ/σ^2 and σ2/λσ^2/λ)? We show that the answer is no, by breaking the direction σ2=O(λ)σ^2=O(λ). What Kahn’s own inequalities give is the following. He proves σ2,λ≤ν−μ≤τ−μ≤κ2/2+O(κ)σ^2,λ≤ν-μ≤τ-μ≤κ^2/2+O(κ), and also κ=O(λ2+λ)κ=O(λ^2+λ) and κ=O(σ4+σ2)κ=O(σ^4+σ^2) (Kahn 2000). Combining these gives σ2=O(λ4) whenever λ=Ω(1),λ≤ν−μ=O(σ8) whenever σ=Ω(1),σ^2=O(λ^4)\ whenever λ= (1), λ≤ν-μ=O(σ^8)\ whenever σ= (1), and Kahn conjectures that the second of these can be improved to ν−μ=O(σ6)ν-μ=O(σ^6), which would be best possible (Kahn 2000). The examples he records all have σ2≍λσ^2 λ. For his Example 7.1, a clique KnK_n with one private pendant leaf at each clique vertex, one has σ2=12n+O(1)σ^2= 12 n+O(1) and λ=κ=n+O(1)λ=κ= n+O(1), and for his Example 7.2 both σ2σ^2 and λ are of order n1/3n^1/3 (Kahn 2000). For a disjoint union of copies of K1,mK_1,m Kahn computes λ=ν−μλ=ν-μ and σ2=m(ν−μ)/(m+1)σ^2=m(ν-μ)/(m+1), so that σ2/λ=m/(m+1)<1σ^2/λ=m/(m+1)<1 (Kahn 2000); there the leaf count moves σ2σ^2 and λ together, and in the construction below it is the clique that decouples them. General criteria for central limit theorems of this kind, in terms of the location of the zeros of graph-counting polynomials, were later given by Lebowitz et al. 2016; they say nothing about the size of λ. Apart from Kahn’s own examples we are aware of no work that bears on Question 7.3. C.7.2 The construction Our graphs are the graphs of Kahn’s Example 7.1 with more leaves. Definition C.7.1. For integers n≥0n≥ 0 and a≥2a≥ 2 let Gn,aG_n,a be the graph with vertex set V(Gn,a)=v1,…,vn∪yi,t:i∈[n],t∈[a−1]V(G_n,a)=\v_1,…,v_n\∪\y_i,t:i∈[n],\ t∈[a-1]\ and edge set E(Gn,a)=vi,vj:1≤i<j≤n∪vi,yi,t:i∈[n],t∈[a−1].E(G_n,a)= \\v_i,v_j\:1≤ i<j≤ n \∪ \\v_i,y_i,t\:i∈[n],\ t∈[a-1] \. That is, Gn,aG_n,a is a clique KnK_n carrying a−1a-1 private pendant leaves at each clique vertex. We call v1,…,vnv_1,…,v_n the clique vertices. The case a=2a=2 is Kahn’s Example 7.1. For n≥1n≥ 1 every leaf has degree 11 and every clique vertex has degree n−1+(a−1)≥1n-1+(a-1)≥ 1, so Gn,aG_n,a is a finite simple graph with δ(Gn,a)≥1δ(G_n,a)≥ 1 and the family satisfies the standing hypothesis of Kahn’s theorem. The parameter a is the number of configurations available at a clique vertex that is not matched inside the clique: it may stay uncovered, or take any one of its a−1a-1 leaves. Figure 9 shows a matching of G5,3G_5,3 and the residual graph it leaves behind. v1v_1v2v_2v3v_3v4v_4v5v_5y3,1y_3,1 Figure 9: The graph G5,3G_5,3, with its ten clique edges and its two private leaves at each clique vertex drawn in gray, and with the vertices covered by the matching drawn solid. The matching M consists of the two heavy black edges v1v2v_1v_2 and v3y3,1v_3y_3,1, so the set of clique vertices missed by the clique part of M is v3,v4,v5\v_3,v_4,v_5\ and exactly one of them takes a leaf. The residual graph FMF_M, the set of edges meeting no edge of M, is drawn heavy and dashed: it is the copy of G2,3G_2,3 carried by the two clique vertices that took no leaf, and ν(FM)=2ν(F_M)=2. Theorem C.7.2. There are absolute constants n0n_0 and c>0c>0 such that for all integers n≥n0n≥ n_0 and a with 4≤a≤n1/44≤ a≤ n^1/4, 14n≤λ(Gn,a)≤ 5n,σ2(Gn,a)≥can. 14 n\ ≤\ λ(G_n,a)\ ≤\ 5 n, σ^2(G_n,a)\ ≥\ c\,a n. In particular σ2(Gn,a)λ(Gn,a)≥c5a. σ^2(G_n,a)λ(G_n,a)\ ≥\ c5\,a. Corollary C.7.3. Put an=⌊n1/4⌋a_n= n^1/4 and Hn=Gn,anH_n=G_n,a_n for n≥16n≥ 16. Then λ(Hn)→∞λ(H_n)→∞, σ2(Hn)→∞σ^2(H_n)→∞, and σ2(Hn)λ(Hn)=Ω(n1/4)⟶∞. σ^2(H_n)λ(H_n)= (n^1/4 ) ∞. Hence there is no absolute constant C with σ2(G)≤Cλ(G)σ^2(G)≤ Cλ(G) for all finite simple graphs G, and λ=Θ(σ2)λ= (σ^2) fails. Both parameters diverge along (Hn)(H_n), so this is not the degenerate kind of counterexample in which σ2σ^2 and λ are both O(1)O(1). Remark C.7.4. Corollary C.7.3 says nothing about the reverse bound λ=O(σ2)λ=O(σ^2), which remains open, and the family (Hn)(H_n) gives no information about it: there λ/σ2→0λ/σ^2→ 0, so the family is consistent with that direction. The best bound we know in that direction is still the one implied by Kahn’s inequalities, namely λ≤ν−μ=O(σ8)λ≤ν-μ=O(σ^8) for σ=Ω(1)σ= (1). Remark C.7.5. The family also limits how much the bound σ2=O(λ4)σ^2=O(λ^4) recorded above can be improved. Indeed an≥12n1/4a_n≥ 12n^1/4, so for n large Theorem C.7.2 gives σ2(Hn)≥12cn3/4σ^2(H_n)≥ 12c\,n^3/4 while λ(Hn)≤5nλ(H_n)≤ 5 n, whence σ2(Hn)≥c2⋅53/2λ(Hn)3/2.σ^2(H_n)\ ≥\ c2· 5^3/2\,λ(H_n)^3/2. Since λ(Hn)→∞λ(H_n)→∞, no bound of the form σ2=O(λβ)σ^2=O(λ^β) can hold for graphs with λ=Ω(1)λ= (1) unless β≥3/2β≥ 3/2. The proof is a direct computation with the exact law of ξGn,a _G_n,a. A matching of Gn,aG_n,a consists of a matching of the clique together with, at each clique vertex left uncovered by it, one of a choices: no leaf, or one of the a−1a-1 leaves. Writing S for the number of clique vertices missed by the clique part of the matching and B for the number of those that do take a leaf, one has ξ=(n−S)/2+Bξ=(n-S)/2+B and ν(FM)=S−Bν(F_M)=S-B, and B is binomial with parameters S and (a−1)/a(a-1)/a. Hence λ=[S]/aλ=E[S]/a, while for a≥4a≥ 4 the variance σ2σ^2 retains a fixed fraction of Var[S]Var[S], and we show that [S]≍anE[S] a n and Var[S]≫anVar[S] a n. The key point is that the leaf count a divides λ but not σ2σ^2: multiplying the number of admissible configurations at a free clique vertex by a shifts the equilibrium of the clique matching, pushing [S]E[S] up to order ana n and Var[S]Var[S] up to at least that order, whereas the residue is governed by the free clique vertices that took no leaf, and their expected number [S]/aE[S]/a stays of order n n. C.7.3 The law of a uniform matching Fix n and a≥2a≥ 2, write G=Gn,aG=G_n,a, and abbreviate ξ=ξG=|M|ξ= _G=|M|. Given a matching M of G, let SM=i∈[n]:vi is not covered by an edge of M∩E(Kn),S=|SM|,S_M=\i∈[n]:\ v_i is not covered by an edge of M∩ E(K_n)\, S=|S_M|, and let B be the number of i∈SMi∈ S_M for which viv_i is matched by M to one of its leaves. Set q=(a−1)/aq=(a-1)/a. Lemma C.7.6. Let M be uniform on ℳ(G)M(G). Then S≡n(mod2)S≡ n 2, and for 0≤s≤n0≤ s≤ n with s≡n(mod2)s≡ n 2, ℙ(S=s)=wsW,ws:=n!s! 2kk!as,k:=n−s2,W:=∑sws.P(S=s)= w_sW, w_s:= n!s!\,2^k\,k!\,a^s, k:= n-s2, W:= _sw_s. (18) Conditionally on S=sS=s, the variable B is binomial Bin(s,q)Bin(s,q). Moreover, for every matching M, |M|=n−S2+B,ν(FM)=S−B.|M|= n-S2+B, ν(F_M)=S-B. (19) Proof. A matching of G is determined by two successive choices: a matching M0⊆E(Kn)M_0 E(K_n) of the clique, and then, for each clique vertex left uncovered by M0M_0, either nothing or one of its a−1a-1 leaves. Indeed leaf edges at distinct clique vertices are disjoint, and a leaf edge at viv_i is compatible with M0M_0 exactly when viv_i is uncovered by M0M_0. The number of matchings M0M_0 of KnK_n leaving a prescribed set of s clique vertices uncovered is the number of perfect matchings of Kn−sK_n-s, namely (n−s)!/(2kk!)(n-s)!/(2^kk!) with k=(n−s)/2k=(n-s)/2; this forces s≡n(mod2)s≡ n 2. Choosing the set of uncovered vertices in (ns) ns ways and then making the leaf choices in asa^s ways gives the number of matchings M with S=sS=s as (ns)(n−s)!2kk!as=n!s! 2kk!as=ws, ns (n-s)!2^kk!\,a^s= n!s!\,2^kk!\,a^s=w_s, which is equation 18. Since the s leaf choices are made independently and uniformly among a options, of which a−1a-1 produce a leaf edge, B is Bin(s,q)Bin(s,q) given S=sS=s. For the first identity in equation 19, note that M consists of (n−S)/2(n-S)/2 clique edges and B leaf edges. For the second, a clique vertex viv_i is uncovered by M precisely when i∈SMi∈ S_M and viv_i took no leaf, which happens for exactly S−BS-B indices, while a leaf yi,ty_i,t is uncovered precisely when it was not chosen. An edge lies in FMF_M if and only if both its ends are uncovered, so FMF_M consists of all clique edges between the S−BS-B uncovered clique vertices together with all a−1a-1 leaf edges at each of them: that is, FM≅GS−B,aF_M G_S-B,\,a. In Gr,aG_r,a with a≥2a≥ 2 the r clique vertices form a vertex cover, so ν≤rν≤ r, while matching each clique vertex to a private leaf shows ν≥rν≥ r. Hence ν(FM)=S−Bν(F_M)=S-B. ∎ Corollary C.7.7. With the notation of Lemma C.7.6, λ(Gn,a)=[S]a,σ2(Gn,a)=(a−22a)2Var[S]+a−1a2[S].λ(G_n,a)= E[S]a, σ^2(G_n,a)= ( a-22a )^2Var[S]+ a-1a^2\,E[S]. (20) Proof. By equation 19 and [B∣S]=qSE[B S]=qS we get λ(Gn,a)=[ν(FM)]=[S−B]=(1−q)[S]=[S]/aλ(G_n,a)=E[ν(F_M)]=E[S-B]=(1-q)E[S]=E[S]/a. Also [ξ∣S]=n2+(q−12)SE[ξ S]= n2+(q- 12)S and Var[ξ∣S]=q(1−q)SVar[ξ S]=q(1-q)S, so decomposing the variance over S, σ2=(q−12)2Var[S]+q(1−q)[S],σ^2= (q- 12 )^2Var[S]+q(1-q)E[S], and q−12=(a−2)/(2a)q- 12=(a-2)/(2a) while q(1−q)=(a−1)/a2q(1-q)=(a-1)/a^2. ∎ Everything therefore reduces to the mean and the variance of the weight sequence equation 18. C.7.4 The distribution of the free set Throughout this subsection and the next we assume 4≤a≤n1/4,L:=an,4≤ a≤ n^1/4, L:=a n, (21) and that n is larger than a suitable absolute constant; all implied constants below are absolute. Since a≤n1/4a≤ n^1/4 gives a3≤ana^3≤ a n, and a≥4a≥ 4, we have a2≤La≤L4,L≤n3/4,L≥4n.a^2≤ La≤ L4, L≤ n^3/4, L≥ 4 n. (22) In particular L→∞L→∞ with n, so every hypothesis of the form “L larger than an absolute constant” below is implied by “n larger than an absolute constant”; we use this without further comment. We also record that 2L+4≤n/22L+4≤ n/2 for n large, by the middle bound in equation 22. From equation 18, for 0≤s≤n−20≤ s≤ n-2 with s≡n(mod2)s≡ n 2, ρ(s):=ws+2ws=a2(n−s)(s+1)(s+2).ρ(s):= w_s+2w_s= a^2(n-s)(s+1)(s+2). (23) The function ρ is strictly decreasing in s, so the sequence (ws)(w_s) is strictly log-concave along the arithmetic progression s≡n(mod2)s≡ n 2, and in particular unimodal. Fix a mode s0s_0, that is, an admissible s0s_0 with ws0=maxswsw_s_0= _sw_s. Lemma C.7.8. Assume equation 21 and n large. Then 12L≤s0≤2L+2 12L≤ s_0≤ 2L+2. Proof. If s≥2Ls≥ 2L is admissible then (s+1)(s+2)>s2≥4L2=4a2n(s+1)(s+2)>s^2≥ 4L^2=4a^2n, so equation 23 gives ρ(s)≤a2ns2≤14.ρ(s)\ ≤\ a^2ns^2\ ≤\ 14. (24) Hence ws+2<wsw_s+2<w_s for every admissible s≥2Ls≥ 2L. If we had s0>2L+2s_0>2L+2 then s0−2≥2Ls_0-2≥ 2L would be admissible and ws0<ws0−2w_s_0<w_s_0-2, contradicting maximality; therefore s0≤2L+2s_0≤ 2L+2. Since s0+2≤2L+4≤ns_0+2≤ 2L+4≤ n, maximality also gives ws0+2≤ws0w_s_0+2≤ w_s_0, that is ρ(s0)≤1ρ(s_0)≤ 1, that is (s0+1)(s0+2)≥a2(n−s0)≥12a2n=12L2,(s_0+1)(s_0+2)\ ≥\ a^2(n-s_0)\ ≥\ 12a^2n= 12L^2, where we used s0≤2L+2≤n/2s_0≤ 2L+2≤ n/2. Hence s0+2≥L/2s_0+2≥ L/ 2, and since L is large this yields s0≥L/2−2≥L/2s_0≥ L/ 2-2≥ L/2. ∎ Lemma C.7.9. Assume equation 21 and n large. Then [S]≤5LE[S]≤ 5L, and consequently λ(Gn,a)≤5nλ(G_n,a)≤ 5 n. Proof. Let T be the least integer with T≥2LT≥ 2L and T≡n(mod2)T≡ n 2, so that T≤2L+2T≤ 2L+2. By equation 24 we have ρ(s)≤14ρ(s)≤ 14 for every admissible s≥Ts≥ T, whence wT+2j≤4−jwTw_T+2j≤ 4^-jw_T for all j≥0j≥ 0. Splitting the expectation at T and using wT≤Ww_T≤ W, [S]≤T+∑j≥0(T+2j)wT+2jW≤T+∑j≥0(T+2j)4−j=73T+89.E[S]\ ≤\ T+ _j≥ 0(T+2j) w_T+2jW\ ≤\ T+ _j≥ 0(T+2j)4^-j= 73T+ 89. Since T≤2L+2T≤ 2L+2 this is at most 143L+509≤5L 143L+ 509≤ 5L, because L is large. Finally λ(Gn,a)=[S]/a≤5L/a=5nλ(G_n,a)=E[S]/a≤ 5L/a=5 n by Corollary C.7.7. ∎ The next lemma is the technical heart of the argument: the distribution of S has no heavy atom. It is what forces Var[S]Var[S] to be large, and it also rules out the degenerate possibility that λ stays bounded. Lemma C.7.10. Assume equation 21 and n large. Then maxsℙ(S=s)≤22L,L=an. _sP(S=s)\ ≤\ 22 L, L=a n. Proof. Put D=⌊L/4⌋D= L/4 ; since L is large we have D≥2D≥ 2 and D≥L/8D≥ L/8. By Lemma C.7.8 and equation 22, s0+2D≤2L+2+L/2≤ns_0+2D≤ 2L+2+ L/2≤ n, so the weights ws0,…,ws0+2Dw_s_0,…,w_s_0+2D are all defined and positive. As s0s_0 is a mode and (ws)(w_s) is unimodal, ws0+2j≥ws0+2Dw_s_0+2j≥ w_s_0+2D for 0≤j≤D0≤ j≤ D, whence W≥∑j=0Dws0+2j≥(D+1)ws0+2DW\ ≥\ _j=0^Dw_s_0+2j\ ≥\ (D+1)\,w_s_0+2D and therefore maxsℙ(S=s)=ws0W≤1D+1⋅ws0ws0+2D=1D+1∏j=0D−1ρ(s0+2j)−1. _sP(S=s)= w_s_0W\ ≤\ 1D+1· w_s_0w_s_0+2D= 1D+1 _j=0^D-1ρ(s_0+2j)^-1. (25) Since ρ is decreasing, ∏j=0D−1ρ(s0+2j)≥ρ(s0+2D)D _j=0^D-1ρ(s_0+2j)≥ρ(s_0+2D)^D, so it remains to bound ρ(s0+2D)ρ(s_0+2D) from below. As s0s_0 is a mode and s0≥12L≥2s_0≥ 12L≥ 2, we have ws0−2≤ws0w_s_0-2≤ w_s_0, that is ρ(s0−2)≥1ρ(s_0-2)≥ 1, which reads a2(n−s0+2)≥(s0−1)s0a^2(n-s_0+2)≥(s_0-1)s_0 and hence a2(n−s0)≥s02−s0−2a2.a^2(n-s_0)\ ≥\ s_0^2-s_0-2a^2. Subtracting 2a2D2a^2D from both sides, a2(n−s0−2D)≥s02−s0−2a2(D+1).a^2(n-s_0-2D)\ ≥\ s_0^2-s_0-2a^2(D+1). Now s0≥L/2s_0≥ L/2 gives s0≤2s02/Ls_0≤ 2s_0^2/L, while a2≤L/4a^2≤ L/4 and D≤L/4D≤ L/4 give 2a2(D+1)≤L2(L4+1)=L3/28+L2≤s02(12L+2L),2a^2(D+1)\ ≤\ L2 ( L4+1 )= L^3/28+ L2\ ≤\ s_0^2 ( 12 L+ 2L ), using s02≥L2/4s_0^2≥ L^2/4. Therefore a2(n−s0−2D)≥s02(1−12L−4L).a^2(n-s_0-2D)\ ≥\ s_0^2 (1- 12 L- 4L ). On the other hand (2D+2)/s0≤(12L+2)/(12L)=1/L+4/L(2D+2)/s_0≤( 12 L+2)/( 12L)=1/ L+4/L, so (s0+2D+1)(s0+2D+2)≤(s0+2D+2)2≤s02(1+1L+4L)2.(s_0+2D+1)(s_0+2D+2)\ ≤\ (s_0+2D+2)^2\ ≤\ s_0^2 (1+ 1 L+ 4L )^2. Combining the last two displays, ρ(s0+2D)≥1−12L−4L(1+1L+4L)2≥ 1−3Lρ(s_0+2D)\ ≥\ 1- 12 L- 4L (1+ 1 L+ 4L )^2\ ≥\ 1- 3 L for L large. Using log(1−x)≥−43x (1-x)≥- 43x for 0≤x≤140≤ x≤ 14 together with D≤L/4D≤ L/4, we get ∏j=0D−1ρ(s0+2j)≥(1−3L)D≥exp(−4DL)≥e−1. _j=0^D-1ρ(s_0+2j)\ ≥\ (1- 3 L )^D\ ≥\ (- 4D L )\ ≥\ e^-1. Substituting into equation 25 and using D+1≥L/8D+1≥ L/8 gives maxsℙ(S=s)≤8e/L≤22/L _sP(S=s)≤ 8e/ L≤ 22/ L. ∎ Lemma C.7.11. Assume equation 21 and n large. Then [S]≥L/4E[S]≥ L/4, and consequently λ(Gn,a)≥n/4λ(G_n,a)≥ n/4. Proof. Let T′T be the largest admissible s with s≤L/2s≤ L/2, so that T′≥L/2−2T ≥ L/2-2. For admissible s≤T′s≤ T we have n−s≥n−L/2≥n/2n-s≥ n-L/2≥ n/2 by equation 22, hence ρ(s)=a2(n−s)(s+1)(s+2)≥a2n/2(L/2+2)2=L2/2(L/2+2)2≥32ρ(s)= a^2(n-s)(s+1)(s+2)\ ≥\ a^2n/2(L/2+2)^2= L^2/2(L/2+2)^2\ ≥\ 32 for L large. Therefore wT′−2j≤(2/3)jwT′w_T -2j≤(2/3)^jw_T for all j≥0j≥ 0, so that ℙ(S≤T′)=∑j≥0wT′−2jW≤ 3wT′W≤ 3maxsℙ(S=s)≤66LP(S≤ T )= _j≥ 0 w_T -2jW\ ≤\ 3\, w_T W\ ≤\ 3 _sP(S=s)\ ≤\ 66 L by Lemma C.7.10. Consequently [S]≥T′ℙ(S>T′)≥(L/2−2)(1−66/L)≥L/4E[S]≥ T P(S>T )≥(L/2-2)(1-66/ L)≥ L/4 for L large. Dividing by a gives λ(Gn,a)=[S]/a≥n/4λ(G_n,a)=E[S]/a≥ n/4. ∎ C.7.5 A variance lower bound We use the following elementary fact, which converts a bound on the largest atom of an integer-valued random variable into a lower bound on its variance. Lemma C.7.12. Let X be an integer-valued random variable and put θ:=maxk∈ℤℙ(X=k)θ:= _k P(X=k). Then Var[X]≥(1−θ)312θ2.Var[X]\ ≥\ (1-θ)^312\,θ^2. Proof. Fix c∈ℝc . For u≥0u≥ 0 the interval [c−u,c+u][c-u,c+u] contains at most 2u+12u+1 integers, so ℙ(|X−c|≤u)≤(2u+1)θP(|X-c|≤ u)≤(2u+1)θ and ℙ(|X−c|>u)≥1−(2u+1)θP(|X-c|>u)≥ 1-(2u+1)θ. Put U=(1−θ)/(2θ)U=(1-θ)/(2θ), the point at which the last expression vanishes, so that the integrand below is nonnegative on [0,U][0,U]. Then [(X−c)2]=∫0∞2uℙ(|X−c|>u)u≥∫0U2u(1−(2u+1)θ)u=U2(1−θ)−43θU3.E [(X-c)^2 ]= _0^∞2u\,P(|X-c|>u)\,du\ ≥\ _0^U2u (1-(2u+1)θ )\,du=U^2(1-θ)- 43θ U^3. Substituting U=(1−θ)/(2θ)U=(1-θ)/(2θ) gives (1−θ)34θ2−(1−θ)36θ2=(1−θ)312θ2, (1-θ)^34θ^2- (1-θ)^36θ^2= (1-θ)^312θ^2, and taking c=[X]c=E[X] proves the claim. ∎ Corollary C.7.13. Assume equation 21 and n large. Then Var[S]≥L/8000Var[S]≥ L/8000. Proof. Write θ=maxsℙ(S=s)θ= _sP(S=s), so θ≤22/Lθ≤ 22/ L by Lemma C.7.10, and θ≤110θ≤ 110 since L is large. The function x↦(1−x)3/(12x2)x (1-x)^3/(12x^2) is decreasing on (0,1)(0,1), so Lemma C.7.12 gives Var[S]≥(1−θ)312θ2≥(9/10)312⋅L484≥L8000.∎Var[S]\ ≥\ (1-θ)^312θ^2\ ≥\ (9/10)^312· L484\ ≥\ L8000. C.7.6 Proof of the main theorem Proof of Theorem C.7.2. Let n0n_0 be an absolute constant large enough for the finitely many hypotheses “n large” invoked above, and assume n≥n0n≥ n_0 and 4≤a≤n1/44≤ a≤ n^1/4. Lemmas C.7.9 and C.7.11 give 14n≤λ(Gn,a)≤5n 14 n≤λ(G_n,a)≤ 5 n. For the variance, a≥4a≥ 4 gives ((a−2)/(2a))2≥(1/4)2=1/16 ((a-2)/(2a) )^2≥(1/4)^2=1/16, so equation 20 and Corollary C.7.13 give σ2(Gn,a)≥116Var[S]≥L128000≥ 10−6an,σ^2(G_n,a)\ ≥\ 116Var[S]\ ≥\ L128000\ ≥\ 10^-6a n, which is the bound on σ2(Gn,a)σ^2(G_n,a) with c=10−6c=10^-6. Dividing the two bounds gives σ2(Gn,a)/λ(Gn,a)≥2⋅10−7aσ^2(G_n,a)/λ(G_n,a)≥ 2· 10^-7a. ∎ Proof of Corollary C.7.3. For n≥maxn0,256n≥ \n_0,256\ the choice a=an=⌊n1/4⌋a=a_n= n^1/4 satisfies 4≤an≤n1/44≤ a_n≤ n^1/4, so Theorem C.7.2 applies to Hn=Gn,anH_n=G_n,a_n. It gives λ(Hn)≥n/4→∞λ(H_n)≥ n/4→∞, then σ2(Hn)≥cann→∞σ^2(H_n)≥ c\,a_n n→∞, and finally σ2(Hn)/λ(Hn)≥can/5=Ω(n1/4)σ^2(H_n)/λ(H_n)≥ c\,a_n/5= (n^1/4). A bound σ2(G)≤Cλ(G)σ^2(G)≤ Cλ(G) valid for all finite simple graphs G would force an≤5C/ca_n≤ 5C/c for every n, which is absurd. ∎ References. Godsil (1981) Chris D Godsil. Matching behaviour is asymptotically normal. Combinatorica, 1(4):369–376, 1981. Heilmann & Lieb (1972) Ole J Heilmann and Elliott H Lieb. Theory of monomer-dimer systems. Communications in mathematical Physics, 25(3):190–232, 1972. Kahn (2000) Jeff Kahn. A normal law for matchings. Combinatorica, 20(3):339–392, 2000. Lebowitz et al. (2016) Joel L Lebowitz, Boris Pittel, David Ruelle, and Eugene R Speer. Central limit theorems, lee–yang zeros, and graph-counting polynomials. Journal of Combinatorial Theory, Series A, 141:147–183, 2016. C.8 An ordered hypergraph extremal function of order nlogn n For a fixed pattern F, Klazar’s extremal function exe(F,n)ex_e(F,n) is the largest number of edges of a simple hypergraph on at most n linearly ordered vertices that contains no order-preserving Berge copy of F. For the five-vertex ordered graph G1G_1 below, the corresponding ordered graph extremal function is Θ(nlogn) (n n), while Klazar’s bounds for the hypergraph function differ by a factor of logn(loglogn)3 n( n)^3. We show that the two functions have the same order of magnitude, so that exe(G1,n)=Θ(nlogn)ex_e(G_1,n)= (n n). The proof compresses a G1G_1-free hypergraph into a single G1G_1-free ordered graph, at the cost of a factor of five and an additive n. C.8.1 Introduction We follow the setup of Klazar 2004a. A hypergraph is a finite list H=(Ej:j∈I)H=(E_j:j∈ I) of finite nonempty subsets of ℕN, called edges. Its vertex set is V(H)=⋃j∈IEjV(H)= _j∈ IE_j, so that H has no isolated vertices, and we write e(H)=|I|e(H)=|I| for the number of edges and i(H)=∑j∈I|Ej|i(H)= _j∈ I|E_j| for the number of vertex–edge incidences. The hypergraph H is simple if its edges are pairwise distinct, and it is a graph if every edge has exactly two elements. Vertices always carry the linear order inherited from ℕN. Given hypergraphs H=(Ej:j∈I)H=(E_j:j∈ I) and F=(Fk:k∈K)F=(F_k:k∈ K), we write H≻FH F, and say that H contains F, if there are an increasing injection ϕ:V(F)→V(H)φ V(F)→ V(H) and an injection ψ:K→Iψ K→ I such that ϕ(Fk)⊆Eψ(k)for every k∈K.φ(F_k) E_ψ(k) every k∈ K. Otherwise H is F-free, written H⊁FH F. Thus a copy of F in H consists of an order-preserving image of the vertices of F together with a choice of pairwise distinct hyperedges of H, one for each edge of F, each containing the image of the edge assigned to it; the chosen hyperedges are allowed to contain further vertices. This is a Berge-type containment that respects the vertex order, and it restricts to ordinary ordered subgraph containment when H and F are both graphs, since then ϕ(Fk)φ(F_k) and Eψ(k)E_ψ(k) both have two elements and the inclusion forces equality. The associated extremal functions are exe(F,n) _e(F,n) =maxe(H):H simple,|V(H)|≤n,H⊁F, = \e(H):\ H simple,\ |V(H)|≤ n,\ H F\, exi(F,n) _i(F,n) =maxi(H):H simple,|V(H)|≤n,H⊁F, = \i(H):\ H simple,\ |V(H)|≤ n,\ H F\, and gex(F,n)gex(F,n) denotes the same maximum of e(G)e(G) taken over simple ordered graphs G alone. Of course gex(F,n)≤exe(F,n)≤exi(F,n)gex(F,n) _e(F,n) _i(F,n) for every F and n. Throughout, G1G_1 denotes the ordered graph G1=(1,3,1,5,2,3,2,4)G_1= (\1,3\,\1,5\,\2,3\,\2,4\ ) on the vertices 1<2<3<4<51<2<3<4<5. Reading 1,2\1,2\ as rows and 3,4,5\3,4,5\ as columns, G1G_1 is the ordered bipartite graph of the 2×32× 3 zero-one matrix (101110) ( smallmatrix1&0&1\\ 1&1&0 smallmatrix ), and Füredi 1990 determined the extremal function of that matrix in the course of bounding the number of unit distances among the vertices of a convex n-gon. The upper bound was obtained independently by Bienstock & Györi 1991. Füredi & Hajnal 1992 began the systematic study of the resulting matrix extremal problems, and Tardos 2019 surveys the ordered graph theory that grew out of them. As Klazar 2004a records, in this particular case the bounds pass from ordered bipartite graphs to all ordered graphs, so that gex(G1,n)=Θ(nlogn).gex(G_1,n)= (n n). (26) Klazar 2004a introduced the containment ≻ and the functions exeex_e and exiex_i, and asked how much of equation 26 survives the passage from ordered graphs to ordered hypergraphs. His Theorem 3.3 gives, in its two parts, nlogn≪exe(G1,n)≤exi(G1,n)≪n(logn)2(loglogn)3,n n _e(G_1,n) _i(G_1,n) n( n)^2( n)^3, the upper bound being obtained by iterating a recursion that controls exe(G1,n)ex_e(G_1,n) in terms of the ordered graph extremal function of blow-ups of G1G_1. He then asked, as Problem 3.4: What is the exact asymptotics of exe(G1,n)ex_e(G_1,n)? We answer this up to the implied constants: the hypergraph function has the same order of magnitude as the graph function. The closest previous work we are aware of on the hypergraph function is Klazar’s own. In a companion paper, Klazar 2004b determines exe(F,n)ex_e(F,n) and exi(F,n)ex_i(F,n) exactly for the 5555 patterns F with at most four incidences, whereas G1G_1 has eight. That paper normalizes by |V(H)|=n|V(H)|=n in place of |V(H)|≤n|V(H)|≤ n; by its Proposition 2.4 the two conventions give the same exe(F,n)ex_e(F,n) unless F consists of distinct singleton edges. Klazar & Marcus 2007 and, independently, Balogh et al. 2006 carry the linear bound of Marcus & Tardos 2004 for excluded permutation matrices from matrices over to ordered hypergraphs, but that theorem applies to permutation patterns, and G1G_1 is not one: by equation 26 its extremal function is already superlinear at the graph level. Theorem C.8.1. For every n≥1n≥ 1, gex(G1,n)≤exe(G1,n)≤n+5gex(G1,n).gex(G_1,n)\ ≤\ ex_e(G_1,n)\ ≤\ n+5gex(G_1,n). In particular exe(G1,n)=Θ(nlogn)ex_e(G_1,n)= (n n). The key point is that a G1G_1-free simple ordered hypergraph is controlled by a single G1G_1-free ordered graph. Process the hyperedges one at a time and record, for each, one pair of its vertices that has not been recorded before. The recorded pairs form an ordered graph B, and a copy of G1G_1 in B pulls back to four distinct hyperedges forming a copy of G1G_1 in the host, so B is G1G_1-free. The hyperedges from which nothing new was recorded are cliques of B, and the crucial observation is that a G1G_1-free ordered graph B has at most 4e(B)4e(B) cliques of size at least two, because in such a graph any edge a,b\a,b\ with a<ba<b has at most two common neighbors to the right of b. C.8.2 Cliques in a G1G_1-free ordered graph We use “clique” to mean any set of pairwise adjacent vertices, not necessarily a maximal one. Lemma C.8.2. Let B be a G1G_1-free simple ordered graph. Then for every edge a,b\a,b\ of B with a<ba<b, the vertices a and b have at most two common neighbors greater than b. Consequently B has at most 4e(B)4e(B) cliques of size at least two. Proof. Fix an edge a,b\a,b\ of B with a<ba<b, and suppose that a and b have three common neighbors greater than b. Choose three such vertices c<d<fc<d<f, so that the five vertices a<b<c<d<fa<b<c<d<f of B carry in particular the four edges a,c,a,f,b,c,b,d.\a,c\, \a,f\, \b,c\, \b,d\. The increasing injection 1↦a1 a, 2↦b2 b, 3↦c3 c, 4↦d4 d, 5↦f5 f carries the four edges of G1G_1 to these four edges of B, so B≻G1B G_1, a contradiction. Now let K be a clique of B with |K|≥2|K|≥ 2 and let a<ba<b be its two smallest vertices. Then a,b\a,b\ is an edge of B, and every remaining vertex of K is a common neighbor of a and b greater than b. Hence K is determined by the edge a,b\a,b\ together with a subset of the set of common neighbors of a and b to the right of b, a set of size at most two. So at most 22=42^2=4 cliques of size at least two have a and b as their two smallest vertices, and summing over the edges of B gives the bound. ∎ C.8.3 Proof of the main theorem Proof of Theorem C.8.1. The lower bound is immediate: a simple ordered graph is a simple ordered hypergraph, and the containment relation ≻ used to define gexgex is the one used to define exeex_e, so the maximum defining exe(G1,n)ex_e(G_1,n) is taken over a larger family. For the upper bound, let H be a simple ordered hypergraph with |V(H)|≤n|V(H)|≤ n and H⊁G1H G_1. Since H is simple, its singleton edges are pairwise distinct one-element subsets of V(H)V(H), so there are at most n of them. We build an ordered graph B from the remaining edges. List the edges of H of size at least two in an arbitrary order and start with B empty. Processing them one at a time, if the current edge E contains a pair a,b\a,b\ that is not yet an edge of B, then add one such pair to B and call E assigned to that pair; otherwise call E unassigned and change nothing. Every assigned edge creates exactly one new edge of B, so assignment is a bijection between the assigned edges of H and the edges of B, and in particular the number of assigned edges is e(B)e(B). Also V(B)⊆V(H)V(B) V(H), so |V(B)|≤n|V(B)|≤ n. We claim that B is G1G_1-free. Suppose instead that there are vertices x1<x2<x3<x4<x5x_1<x_2<x_3<x_4<x_5 of B such that x1,x3,x1,x5,x2,x3,x2,x4\x_1,x_3\, \x_1,x_5\, \x_2,x_3\, \x_2,x_4\ are all edges of B. These are four distinct edges of B, so the hyperedges assigned to them are four distinct edges D1,D2,D3,D4D_1,D_2,D_3,D_4 of H with x1,x3⊆D1,x1,x5⊆D2,x2,x3⊆D3,x2,x4⊆D4.\x_1,x_3\ D_1, \x_1,x_5\ D_2, \x_2,x_3\ D_3, \x_2,x_4\ D_4. Taking ϕ(i)=xiφ(i)=x_i for 1≤i≤51≤ i≤ 5 and sending the edges 1,3,1,5,2,3,2,4\1,3\,\1,5\,\2,3\,\2,4\ of G1G_1 to D1,D2,D3,D4D_1,D_2,D_3,D_4 respectively gives H≻G1H G_1, contrary to hypothesis. Hence B is G1G_1-free, and therefore e(B)≤gex(G1,n).e(B) (G_1,n). It remains to count the unassigned edges. If an edge E of H with |E|≥2|E|≥ 2 is unassigned, then at the moment E was processed every pair contained in E was already an edge of B; since edges are only ever added to B, the same holds at the end of the process, so E is a clique of B of size at least two. Distinct edges of H are distinct sets because H is simple, so distinct unassigned edges give distinct cliques of B. By Lemma C.8.2 the number of unassigned edges is therefore at most 4e(B)4e(B). Collecting the three types of edge, e(H)≤n+e(B)+4e(B)≤n+5gex(G1,n),e(H)\ ≤\ n+e(B)+4e(B)\ ≤\ n+5gex(G_1,n), which is the upper bound. The final assertion follows by combining the two bounds with equation 26. ∎ Remark C.8.3. The argument above bounds only the number of edges, since the edges assigned in the auxiliary construction may have large cardinality. The incidence count is nevertheless controlled by Klazar 2004a, which states that exi(F,n)≤(2p−1)(q−1)exe(F,n)ex_i(F,n)≤(2p-1)(q-1)ex_e(F,n) whenever F has no two separated edges, where p=|V(F)|p=|V(F)| and q=e(F)>1q=e(F)>1; here two edges are separated if every vertex of the first precedes every vertex of the second. Every edge of G1G_1 has largest vertex at least 33 and smallest vertex at most 22, so no edge of G1G_1 lies entirely to the left of another and the hypothesis holds; with p=5p=5 and q=4q=4 it gives exi(G1,n)≤27exe(G1,n)ex_i(G_1,n)≤ 27ex_e(G_1,n). This is Klazar’s own route from part 1 to part 2 of his Theorem 3.3, and applied to Theorem C.8.1 it yields exi(G1,n)=Θ(nlogn)ex_i(G_1,n)= (n n) as well. References. Balogh et al. (2006) József Balogh, Béla Bollobás, and Robert Morris. Hereditary properties of partitions, ordered graphs and ordered hypergraphs. European Journal of combinatorics, 27(8):1263–1281, 2006. Bienstock & Györi (1991) Dan Bienstock and Ervin Györi. An extremal problem on sparse 0-1 matrices. SIAM Journal on Discrete Mathematics, 4(1):17–27, 1991. Füredi (1990) Zoltán Füredi. The maximum number of unit distances in a convex n-gon. Journal of Combinatorial Theory, Series A, 55(2):316–320, 1990. Füredi & Hajnal (1992) Zoltán Füredi and Péter Hajnal. Davenport-schinzel theory of matrices. Discrete Mathematics, 103(3):233–251, 1992. Klazar (2004a) Martin Klazar. Extremal problems for ordered (hyper) graphs: applications of davenport–schinzel sequences. European Journal of Combinatorics, 25(1):125–140, 2004a. Klazar (2004b) Martin Klazar. Extremal problems for ordered hypergraphs: small patterns and some enumeration. Discrete applied mathematics, 143(1-3):144–154, 2004b. Klazar & Marcus (2007) Martin Klazar and Adam Marcus. Extensions of the linear bound in the füredi–hajnal conjecture. Advances in Applied Mathematics, 38(2):258–266, 2007. Marcus & Tardos (2004) Adam Marcus and Gábor Tardos. Excluded permutation matrices and the stanley–wilf conjecture. Journal of Combinatorial Theory, Series A, 107(1):153–160, 2004. Tardos (2019) Gábor Tardos. Extremal theory of vertex or edge ordered graphs1. Surveys in combinatorics 2019, 456:221, 2019. C.9 Eigenvalues below −2-2 under repeated subdivision This problem concerns the eigenvalues that a graph retains below −2-2 when a fixed set of its edges is subdivided over and over. We show that their number is eventually constant, and that the constant is the number of negative eigenvalues of a single fixed matrix on the original vertex set, namely 2In+AR−DS2I_n+A_R-D_S, where ARA_R records the edges that are never subdivided and DSD_S the degrees in the edges that are. The proof eliminates the subdivision vertices by a Schur complement, which is legitimate because the block they contribute is positive definite, and then compares the resulting n×n× n matrices in the Loewner order. C.9.1 Introduction Let G be a finite simple graph on n=|V(G)|n=|V(G)| vertices, let S⊆E(G)S E(G) be a fixed set of edges, and write R=E(G)∖SR=E(G) S for the rest. For t≥1t≥ 1 let Gt=Gt(S)G_t=G_t(S) be the graph obtained from G by replacing every edge uv∈Suv∈ S with a path of length t from u to v, the t-stretch of uvuv, whose t−1t-1 internal vertices are new and lie on no other stretch. The edges of R are left untouched, and G1=G_1=G. We write λ1(X)≥⋯≥λ|V(X)|(X) _1(X)≥·s≥ _|V(X)|(X) for the adjacency eigenvalues of a graph X and mX(a,b):=#i:λi(X)∈(a,b)m_X(a,b):=\#\i: _i(X)∈(a,b)\ for the number of them in an interval, counted with multiplicity. For a real symmetric matrix M we write n−(M)n_-(M) for its number of negative eigenvalues, counted with multiplicity, that is, for its negative index of inertia. Finally ARA_R and ASA_S denote the adjacency matrices of the spanning subgraphs (V(G),R)(V(G),R) and (V(G),S)(V(G),S), and DSD_S the diagonal matrix of the degrees degS(v) _S(v) in (V(G),S)(V(G),S), all indexed by V(G)V(G); the degree of v in (V(G),R)(V(G),R) is written degR(v) _R(v). The window [−2,2][-2,2] is the one that matters here. Deleting Q=v∈V(G):degG(v)≥3Q=\v∈ V(G): _G(v)≥ 3\ from GtG_t leaves a disjoint union of paths and cycles, whose eigenvalues all lie in [−2,2][-2,2], so interlacing bounds the number of eigenvalues of GtG_t above 22, and likewise the number below −2-2, by |Q||Q|, uniformly in t (Kumar et al. 2025). The value −2-2 is also the classical threshold on the negative side: a connected graph with mG(−∞,−2)=0m_G(-∞,-2)=0 is a generalized line graph or one of finitely many exceptional graphs representable in the root system E8E_8, by the classification of Cameron et al. 1991, and the structure theory of graphs with least eigenvalue −2-2 is built around that dichotomy (Cvetkovic et al. 2004). So mGt(−∞,−2)m_G_t(-∞,-2) measures how far the subdivided graph is from being a generalized line graph, and the question is whether repeated subdivision settles this quantity down. Subdivision has been studied spectrally since the theorem of Hoffman & Smith 1974 on the spectral radii of topologically equivalent graphs, and it is the basic operation in Hoffman’s program on limit points of spectral radii, surveyed by Wang et al. 2025. Subdividing a subset of the edges is exactly the operation used by Haiman et al. 2022 to construct graphs with high approximate second eigenvalue multiplicity, showing that the multiplicity bound of Jiang et al. 2021 behind the resolution of the equiangular lines problem is sharp in that relaxed sense. Motivated by this, Kumar et al. 2025 analyze the whole spectrum of GtG_t as t→∞t→∞. They study the four sequences mGt(2,∞)m_G_t(2,∞), mGt(−∞,−2)m_G_t(-∞,-2), mHt(2,∞)m_H_t(2,∞), and mHt(−∞,−2)m_H_t(-∞,-2), where Ht=Ht(S)H_t=H_t(S) is obtained from G2t+1(S)G_2t+1(S) by deleting the middle edge of every stretch. Since HtH_t is an induced subgraph of Ht+1H_t+1, the two H-sequences are nondecreasing, and subdividing a single edge cannot decrease the number of eigenvalues above 22, so mGt(2,∞)m_G_t(2,∞) is nondecreasing as well. Being nondecreasing and bounded by |Q||Q|, those three sequences are eventually constant. The fourth sequence is not monotone, and is left open as Kumar et al. 2025: There exists t0∈ℕt_0 such that mGt(−∞,−2)m_G_t(-∞,-2) is constant for all t≥t0t≥ t_0. The failure of monotonicity is genuine and is caused by parity: subdividing changes the length of every cycle through a stretch, and Figure 1 of Kumar et al. 2025 exhibits a graph whose successive subdivisions have 11, 00, and 11 eigenvalues below −2-2. Example C.9.4 below returns to that graph and follows the sequence to the end. In the special case S=E(G)S=E(G) the conjecture already follows from the same paper. There Kumar et al. 2025 show that for each fixed k≤nk≤ n the eigenvalue λ|V(Gt)|−k+1(Gt) _|V(G_t)|-k+1(G_t) tends to −dk/dk−1-d_k/ d_k-1 when dk≥3d_k≥ 3 and to −2-2 otherwise, where d1≥⋯≥dnd_1≥·s≥ d_n is the degree sequence of G, so at least |Q||Q| eigenvalues of GtG_t lie below −2-2 once t is large, while their interlacing bound caps the count at |Q||Q| for every t. For a general S the upper half of this argument survives, since that cap is proved for an arbitrary S; the lower half does not. They do prove that each individual eigenvalue λ|V(Gt)|−k+1(Gt) _|V(G_t)|-k+1(G_t) converges, but for a general S the limit is not known in closed form and may be −2-2 itself, and an eigenvalue converging to −2-2 is free to cross the threshold again and again. That boundary case is the whole difficulty. Since the count never exceeds |Q||Q|, it equals the number of k≤|Q|k≤|Q| with λ|V(Gt)|−k+1(Gt)<−2 _|V(G_t)|-k+1(G_t)<-2; the indices whose limit differs from −2-2 contribute a constant from some point on, and Conjecture 20 asserts that the remaining ones do too. We are aware of no work on the conjecture itself; the papers citing Kumar et al. 2025 pursue other lines, among them the Hoffman program (Wang et al. 2025) and the effect of subdivision on other spectral parameters, such as the Perron eigenvalue of the Ricci matrix of a tree (Bai et al. 2026), which is also handled by eliminating the internal structure with a Schur complement. We prove the conjecture for every G and every S, and identify the constant. Theorem C.9.1. Let G be a finite simple graph on n vertices, let S⊆E(G)S E(G), let R=E(G)∖SR=E(G) S, and put K∞:=2In+AR−DS.K_∞:=2I_n+A_R-D_S. (27) Then mGt(−∞,−2)≤n−(K∞)for every t≥2,m_G_t(-∞,-2)≤ n_-(K_∞) every t≥ 2, with equality for all sufficiently large t. In particular mGt(−∞,−2)m_G_t(-∞,-2) is eventually constant, and its eventual value is n−(K∞)n_-(K_∞). When every edge is subdivided the matrix K∞K_∞ is diagonal and the eventual value can be read off the degree sequence. Corollary C.9.2. If S=E(G)S=E(G), then mGt(−∞,−2)=|Q|m_G_t(-∞,-2)=|Q| for all sufficiently large t, where Q=v∈V(G):degG(v)≥3Q=\v∈ V(G): _G(v)≥ 3\. This is the value predicted by Kumar et al. 2025, and it makes explicit the negative-side case of the tightness they assert for their interlacing bound. Remark C.9.3. The threshold in Theorem C.9.1 is effective. Write r=n−(K∞)r=n_-(K_∞), and note that for r=0r=0 the upper bound already forces equality for every t≥2t≥ 2. For r≥1r≥ 1 let γ:=−λn−r+1(K∞)>0γ:=- _n-r+1(K_∞)>0 be the smallest absolute value of a negative eigenvalue of K∞K_∞. The proof gives equality for every t≥2t≥ 2 with t>2Δ(G)/γt>2 (G)/γ, where Δ(G) (G) is the maximum degree of G. Moreover K∞K_∞ has integer entries and spectral radius at most 2+Δ(G)2+ (G), which forces γ≥(2+Δ(G))−(n−1)γ≥(2+ (G))^-(n-1). Hence equality holds for every t≥2t≥ 2 with t>2Δ(G)(2+Δ(G))n−1t>2 (G)(2+ (G))^n-1. Example C.9.4. Let G be the cycle v1v2v3v4v_1v_2v_3v_4 together with a pendant edge v1v5v_1v_5, and let S=v2v3S=\v_2v_3\, so that GtG_t is the cycle Ct+3C_t+3 with a pendant edge attached. This is the family in Figure 1 of Kumar et al. 2025, whose first three members have 11, 00, and 11 eigenvalues below −2-2. Here DS=diag(0,1,1,0,0)D_S=diag(0,1,1,0,0) and K∞=(2101111000001101012010002),K_∞= pmatrix2&1&0&1&1\\ 1&1&0&0&0\\ 0&0&1&1&0\\ 1&0&1&2&0\\ 1&0&0&0&2 pmatrix, whose eigenvalues are approximately −0.1388-0.1388, 0.54010.5401, 1.51021.5102, 2.38352.3835 and 3.70503.7050, so n−(K∞)=1n_-(K_∞)=1. Theorem C.9.1 therefore predicts the value 11, and indeed the sequence mGt(−∞,−2)m_G_t(-∞,-2) for t=1,2,3,…t=1,2,3,… is 1, 0, 1, 0, 1, 1, 1, 1,…,1,\;0,\;1,\;0,\;1,\;1,\;1,\;1,\;…, oscillating with the parity of the cycle length until t=5t=5 and constant thereafter. The effective threshold of Remark C.9.3 gives only t>2⋅3/γt>2· 3/γ with γ≈0.1388γ≈ 0.1388, that is t≥44t≥ 44, so the bound is far from sharp on this example. The proof has two steps. The first is exact and holds for every t≥2t≥ 2: eliminating the subdivision vertices replaces A(Gt)+2IA(G_t)+2I with an n×n× n matrix Kt=K∞+1t(DS−(−1)tAS)K_t=K_∞+ 1t (D_S-(-1)^tA_S ) (28) of the same negative inertia. The second is a comparison: the perturbation in equation 28 is positive semidefinite for every parity of t, being the Laplacian of (V(G),S)(V(G),S) for even t and its signless Laplacian for odd t, and it tends to 00. So KtK_t approaches K∞K_∞ from above in the Loewner order, which pins the negative inertia from both sides. C.9.2 Eliminating the subdivision vertices Write PℓP_ for the path on ℓ vertices and put Tℓ:=2Iℓ+A(Pℓ)T_ :=2I_ +A(P_ ). Lemma C.9.5. For every ℓ≥1 ≥ 1 the matrix TℓT_ is positive definite, detTℓ=ℓ+1 T_ = +1, and (Tℓ−1)1,1=(Tℓ−1)ℓ,ℓ=ℓ+1,(Tℓ−1)1,ℓ=(Tℓ−1)ℓ,1=(−1)ℓ+1ℓ+1.(T_ ^-1)_1,1=(T_ ^-1)_ , = +1, (T_ ^-1)_1, =(T_ ^-1)_ ,1= (-1) +1 +1. Proof. For y=(y1,…,yℓ)∈ℝℓy=(y_1,…,y_ ) T , yTℓy=2∑i=1ℓyi2+2∑i=1ℓ−1yiyi+1=y12+yℓ2+∑i=1ℓ−1(yi+yi+1)2.y TT_ y=2 _i=1 y_i^2+2 _i=1 -1y_iy_i+1=y_1^2+y_ ^2+ _i=1 -1(y_i+y_i+1)^2. If this vanishes then y1=0y_1=0 and yi+1=−yiy_i+1=-y_i for all i, so y=0y=0; hence TℓT_ is positive definite. Expanding detTℓ T_ along the first row gives detTℓ=2detTℓ−1−detTℓ−2 T_ =2 T_ -1- T_ -2 with detT0=1 T_0=1 and detT1=2 T_1=2, whence detTℓ=ℓ+1 T_ = +1. By Cramer’s rule, (Tℓ−1)1,1(T_ ^-1)_1,1 is the determinant of the matrix obtained by deleting the first row and column, namely detTℓ−1=ℓ T_ -1= , divided by detTℓ=ℓ+1 T_ = +1; the entry (Tℓ−1)ℓ,ℓ(T_ ^-1)_ , is the same by symmetry. For the corner entry, (Tℓ−1)1,ℓ=(−1)ℓ+1det(N′)/detTℓ(T_ ^-1)_1, =(-1) +1 (N )/ T_ , where N′N is TℓT_ with its last row and first column deleted. The rows of N′N are indexed by 1≤i≤ℓ−11≤ i≤ -1 and its columns by 2≤j≤ℓ2≤ j≤ , and the entry in position (i,j)(i,j) vanishes unless |i−j|≤1|i-j|≤ 1; reindexing the columns by k=j−1k=j-1 makes N′N lower triangular with every diagonal entry equal to (Tℓ)i,i+1=1(T_ )_i,i+1=1. Hence detN′=1 N =1 and (Tℓ−1)1,ℓ=(−1)ℓ+1/(ℓ+1)(T_ ^-1)_1, =(-1) +1/( +1). ∎ uuw1w_1w2w_2⋯·swt−1w_t-1vvTt−1T_t-1uuvv−(−1)t/t-(-1)^t/t−(t−1)/t-(t-1)/t−(t−1)/t-(t-1)/t Figure 10: The t-stretch of an edge uv∈Suv∈ S inside GtG_t, on the left, and its contribution to the Schur complement, on the right. The t−1t-1 internal vertices, drawn hollow, carry the positive definite block Tt−1T_t-1 of A(Gt)+2IA(G_t)+2I; eliminating them subtracts (t−1)/t(t-1)/t from each of the diagonal entries at u and at v and subtracts (−1)t/t(-1)^t/t from the entries at uvuv and at vuvu, which is the content of Proposition C.9.6. Proposition C.9.6. For every t≥2t≥ 2, mGt(−∞,−2)=n−(Kt),Kt:=2In+AR−t−1tDS−(−1)tAS,m_G_t(-∞,-2)=n_-(K_t), K_t:=2I_n+A_R- t-1tD_S- (-1)^ttA_S, and KtK_t satisfies equation 28. Proof. Put Bt:=A(Gt)+2IB_t:=A(G_t)+2I. An eigenvalue λ of A(Gt)A(G_t) satisfies λ<−2λ<-2 exactly when λ+2<0λ+2<0, so mGt(−∞,−2)=n−(Bt).m_G_t(-∞,-2)=n_-(B_t). Partition V(Gt)V(G_t) into the original vertices V=V(G)V=V(G) and the set WtW_t of internal vertices of the stretches. Two internal vertices are adjacent in GtG_t only if they lie on the same stretch and are consecutive on it, so Bt[Wt,Wt]B_t[W_t,W_t] is block diagonal with one block Tt−1T_t-1 for each edge of S, as in Figure 10. By Lemma C.9.5 this block is positive definite, so Haynsworth’s inertia additivity formula (Haynsworth 1968) applies and gives n−(Bt)=n−(Bt[V,V]−Bt[V,Wt]Bt[Wt,Wt]−1Bt[Wt,V]).n_-(B_t)=n_- (B_t[V,V]-B_t[V,W_t]\,B_t[W_t,W_t]^-1\,B_t[W_t,V] ). It remains to identify the Schur complement on the right. No edge of S survives in GtG_t for t≥2t≥ 2, so Bt[V,V]=2In+ARB_t[V,V]=2I_n+A_R. Label the stretch of uv∈Suv∈ S as u,w1,…,wℓ,vu,w_1,…,w_ ,v with ℓ=t−1 =t-1, so that the column of Bt[V,Wt]B_t[V,W_t] at w1w_1 is eue_u, the column at wℓw_ is eve_v, and all other columns of that stretch vanish; here eu,ev∈ℝVe_u,e_v ^V are standard basis vectors, and for ℓ=1 =1 the single column is eu+eve_u+e_v. Writing C=Tℓ−1C=T_ ^-1, the stretch of uvuv therefore contributes C1,1eueu+Cℓ,ℓevev+C1,ℓ(euev+eveu)C_1,1e_ue_u T+C_ , e_ve_v T+C_1, (e_ue_v T+e_ve_u T ) to Bt[V,Wt]Bt[Wt,Wt]−1Bt[Wt,V]B_t[V,W_t]B_t[W_t,W_t]^-1B_t[W_t,V], and this formula is correct for ℓ=1 =1 as well, since then C1,1=Cℓ,ℓ=C1,ℓ=12C_1,1=C_ , =C_1, = 12. By Lemma C.9.5 with ℓ=t−1 =t-1 we have C1,1=Cℓ,ℓ=(t−1)/tC_1,1=C_ , =(t-1)/t and C1,ℓ=(−1)t/tC_1, =(-1)^t/t. Summing over S and using ∑uv∈S(eueu+evev)=DS _uv∈ S(e_ue_u T+e_ve_v T)=D_S and ∑uv∈S(euev+eveu)=AS _uv∈ S(e_ue_v T+e_ve_u T)=A_S, the Schur complement equals KtK_t as displayed. Finally Kt=2In+AR−DS+1tDS−(−1)tAS=K∞+1t(DS−(−1)tAS),K_t=2I_n+A_R-D_S+ 1tD_S- (-1)^ttA_S=K_∞+ 1t (D_S-(-1)^tA_S ), which is equation 28. ∎ Remark C.9.7. For t=2t=2 and S=E(G)S=E(G) the graph G2G_2 is the subdivision graph of G and Proposition C.9.6 reads K2=12(4In−(D+A(G)))K_2= 12 (4I_n-(D+A(G)) ), where D is the degree matrix of G and D+A(G)D+A(G) is the signless Laplacian of G. So mG2(−∞,−2)m_G_2(-∞,-2) is the number of signless Laplacian eigenvalues of G exceeding 44. Indeed, if N is the vertex-edge incidence matrix of G, then in the block form given by V(G)V(G) and the subdivision vertices A(G2)=(0N0),A(G_2)= pmatrix0&N\\ N T&0 pmatrix, so the nonzero eigenvalues of A(G2)A(G_2) are the numbers ±q± q with q a nonzero eigenvalue of NN=D+A(G)N T=D+A(G), and −q<−2- q<-2 exactly when q>4q>4. C.9.3 Comparison in the Loewner order Lemma C.9.8. For every t≥2t≥ 2 we have Kt⪰K∞K_t K_∞ in the Loewner order and ‖Kt−K∞‖≤2Δ(G)/t\|K_t-K_∞\|≤ 2 (G)/t, where ∥⋅∥\|·\| is the spectral norm. Proof. Fix an arbitrary orientation of each edge of S. For x∈ℝV(G)x ^V(G), x(DS−(−1)tAS)x=∑uv∈S(xu−(−1)txv)2≥0,x T (D_S-(-1)^tA_S )x= _uv∈ S (x_u-(-1)^tx_v )^2≥ 0, since expanding the squares returns xDSx−(−1)txASx TD_Sx-(-1)^tx TA_Sx and the summands do not depend on the chosen orientation. Thus DS−(−1)tASD_S-(-1)^tA_S is positive semidefinite, being the Laplacian of (V(G),S)(V(G),S) for even t and its signless Laplacian for odd t, and equation 28 gives Kt−K∞=1t(DS−(−1)tAS)⪰0K_t-K_∞= 1t (D_S-(-1)^tA_S ) 0. For the norm bound, the spectral norm of a real symmetric matrix is at most its largest absolute row sum, and the row of DS−(−1)tASD_S-(-1)^tA_S indexed by v has diagonal entry degS(v) _S(v) and a further degS(v) _S(v) entries of absolute value 11, one for each S-neighbor of v, so that row sum is 2degS(v)≤2Δ(G)2 _S(v)≤ 2 (G), and the factor 1/t1/t yields the stated bound. ∎ C.9.4 Proof of the main theorem Proof of Theorem C.9.1. Put r=n−(K∞)r=n_-(K_∞) and fix t≥2t≥ 2. By Proposition C.9.6 it suffices to prove that n−(Kt)≤rn_-(K_t)≤ r always, and that n−(Kt)≥rn_-(K_t)≥ r once t is large. For the upper bound we may assume r<nr<n, since otherwise there is nothing to prove. By definition of r we have λn−r(K∞)≥0 _n-r(K_∞)≥ 0, and Kt⪰K∞K_t K_∞ gives λj(Kt)≥λj(K∞) _j(K_t)≥ _j(K_∞) for every j by Weyl monotonicity (Horn & Johnson 2012). Hence λn−r(Kt)≥0 _n-r(K_t)≥ 0, so at most r eigenvalues of KtK_t are negative, that is, n−(Kt)≤rn_-(K_t)≤ r. This holds for every t≥2t≥ 2 and is the asserted inequality. For the lower bound we may assume r≥1r≥ 1, since for r=0r=0 the upper bound already gives n−(Kt)=0n_-(K_t)=0 for every t≥2t≥ 2. Let γ:=−λn−r+1(K∞)>0γ:=- _n-r+1(K_∞)>0 be the gap of Remark C.9.3, so that λj(K∞)≤−γ _j(K_∞)≤-γ for every j≥n−r+1j≥ n-r+1. Weyl’s perturbation inequality together with Lemma C.9.8 gives λj(Kt)≤λj(K∞)+‖Kt−K∞‖≤−γ+2Δ(G)t _j(K_t)≤ _j(K_∞)+\|K_t-K_∞\|≤-γ+ 2 (G)t for those j, which is negative as soon as t>2Δ(G)/γt>2 (G)/γ. For such t at least r eigenvalues of KtK_t are negative, so n−(Kt)≥rn_-(K_t)≥ r. Combining the two bounds, n−(Kt)=rn_-(K_t)=r for every t>max2,2Δ(G)/γt> \2,2 (G)/γ\, and Proposition C.9.6 turns this into mGt(−∞,−2)=r=n−(K∞)m_G_t(-∞,-2)=r=n_-(K_∞), as desired. ∎ Proof of Remark C.9.3. The first assertion is the threshold t>2Δ(G)/γt>2 (G)/γ displayed in the proof above. For the second, the spectral norm of K∞K_∞ is at most its largest absolute row sum, which is maxv(|2−degS(v)|+degR(v))≤2+Δ(G) _v (|2- _S(v)|+ _R(v) )≤ 2+ (G), so every eigenvalue of K∞K_∞ has absolute value at most 2+Δ(G)2+ (G). On the other hand K∞K_∞ has integer entries, so the product of its nonzero eigenvalues is, up to sign, the last nonvanishing coefficient of its characteristic polynomial and hence a nonzero integer. The absolute values of those at most n eigenvalues therefore multiply to at least 11, and each factor is at most 2+Δ(G)2+ (G), so the smallest of them, and in particular γ, is at least (2+Δ(G))−(n−1)(2+ (G))^-(n-1). ∎ Proof of Corollary C.9.2. When S=E(G)S=E(G) we have AR=0A_R=0 and DSD_S equal to the degree matrix D of G, so equation 27 makes K∞=2In−DK_∞=2I_n-D diagonal with entries 2−degG(v)2- _G(v). Such an entry is negative exactly when degG(v)≥3 _G(v)≥ 3, so n−(K∞)=|Q|n_-(K_∞)=|Q|. ∎ References. Bai et al. (2026) Shuliang Bai, Haoxuan Cheng, and Bobo Hua. Edge subdivision and the perron eigenvalue of tree ricci matrices. arXiv preprint arXiv:2605.30949, 2026. Cameron et al. (1991) Peter J Cameron, Jean-Marie Goethals, Johan Jacob Seidel, and Ernest E Shult. Line graphs, root systems, and elliptic geometry. In Geometry and Combinatorics, p. 208–230. Elsevier, 1991. Cvetkovic et al. (2004) Dragoš Cvetkovic, Peter Rowlinson, and Slobodan Simic. Spectral generalizations of line graphs: On graphs with least eigenvalue-2, volume 314. Cambridge University Press, 2004. Haiman et al. (2022) Milan Haiman, Carl Schildkraut, Shengtong Zhang, and Yufei Zhao. Graphs with high second eigenvalue multiplicity. Bulletin of the London Mathematical Society, 54(5):1630–1652, 2022. Haynsworth (1968) Emilie V Haynsworth. Determination of the inertia of a partitioned hermitian matrix. Linear algebra and its applications, 1(1):73–81, 1968. Hoffman & Smith (1974) Alan J Hoffman and John Howard Smith. On the spectral radii of topologically equivalent graphs. IBM Thomas J. Watson Research Division, 1974. Horn & Johnson (2012) Roger A Horn and Charles R Johnson. Matrix analysis. Cambridge university press, 2012. Jiang et al. (2021) Zilin Jiang, Jonathan Tidor, Yuan Yao, Shengtong Zhang, and Yufei Zhao. Equiangular lines with a fixed angle. Annals of Mathematics, 194(3):729–743, 2021. Kumar et al. (2025) Hitesh Kumar, Bojan Mohar, Shivaramakrishna Pragada, and Hanmeng Zhan. Subdivision and graph eigenvalues. Linear Algebra and its Applications, 710:336–355, 2025. Wang et al. (2025) Jianfeng Wang, Jing Wang, Maurizio Brunetti, Francesco Belardo, and Ligong Wang. Developments on the hoffman program of graphs. Advances in Applied Mathematics, 169:102915, 2025. C.10 Small unions of lines closing a route to the Nikodym bound Lund et al. 2018 conjectured that a family of Ω(q3) (q^3) lines in q3F_q^3, no plane of which contains a superlinear number of them, must cover all but a vanishing proportion of q3F_q^3. We show that the union can have density 1/2+o(1)1/2+o(1). For every odd prime power q there is a family L of q2(q+1)/2q^2(q+1)/2 lines, at most q+1q+1 of which lie in any single plane, whose union has exactly q2(q+1)/2q^2(q+1)/2 points. The family consists of one half of the tangent lines to the paraboloid z=x2−νy2z=x^2-ν y^2, where ν is a fixed nonsquare, the half being selected by the quadratic character of the direction. The same family also refutes the sharper quantitative conjecture that they state alongside the first one. C.10.1 Introduction For a set L of affine lines in q3F_q^3, write P(L)=⋃ℓ∈LℓP(L)= _ ∈ L for the set of points lying on some line of L. The problem is to bound |P(L)||P(L)| from below when L is large and no plane of q3F_q^3 carries too much of it. In Section 1.2.1 of their paper on Kakeya and Nikodym sets in three dimensions, Lund et al. 2018 propose the following, their Conjecture 1.4. Let C>0C>0 be a constant independent of q, and let α(q)∈ω(q)α(q)∈ω(q). If L is a set of at least Cq3Cq^3 lines and no plane contains α(q)α(q) lines of L, then |P(L)|≥(1−o(1))q3|P(L)|≥(1-o(1))q^3. The o(1)o(1) is a function of q that depends on C and α. In the same place they propose the following sharper form, their Conjecture 1.5. Let ε>0 >0 be any constant and let q be a sufficiently large prime power. Let L be a set of at least q5/2+εq^5/2+ lines in q3F_q^3 such that no plane contains more than (1/2)q3/2(1/2)q^3/2 lines of L. Then, |P(L)|≥q3−O(q5/2)|P(L)|≥ q^3-O(q^5/2). These two conjectures are the geometric heart of Lund et al. 2018. For q sufficiently large, their Theorem 1.1 gives |K|≥0.2107q3|K|≥ 0.2107q^3 for every Kakeya set K⊆q3K _q^3 and their Theorem 1.3 gives ||≥0.38q3|N|≥ 0.38q^3 for every Nikodym set ⊆q3N _q^3, while their Theorem 3.8 shows that Conjecture 1.4 would upgrade the second of these to the conjectured optimal bound (1−o(1))q3(1-o(1))q^3. Conjecture 1.5 is carefully calibrated against two constructions of Lund et al. 2018: from a nondegenerate Hermitian variety, for square q=p2q=p^2 and each 0<θ<10<θ<1, they produce a set L of (θ+o(1))q7/2(θ+o(1))q^7/2 lines with no plane containing more than (θ+o(1))q3/2(θ+o(1))q^3/2 of them and with |P(L)|≤q3−(1−θ+o(1))q5/2|P(L)|≤ q^3-(1-θ+o(1))q^5/2, so the error term O(q5/2)O(q^5/2) in the conclusion of Conjecture 1.5 cannot be improved; a second construction of theirs, the union of O(q1/2)O(q^1/2) Hermitian varieties, shows that the hypothesis |L|≥q5/2+ε|L|≥ q^5/2+ cannot be substantially relaxed. They also prove that any 0.62q30.62q^3 lines in q3F_q^3 already satisfy |P(L)|≥(0.38−o(1))q3|P(L)|≥(0.38-o(1))q^3 with no hypothesis on planes at all, and that without such a hypothesis this is close to sharp: for large q, they take all lines lying in a union of 0.62q0.62q planes through a common point and obtain (1−o(1))0.62q3(1-o(1))0.62q^3 lines whose union has fewer than 0.43q30.43q^3 points. It is this flat example that the plane hypothesis of Conjecture 1.4 is designed to exclude, and our family shows that excluding it does not help: the half-tangent family has at most q+1q+1 lines in any plane, against order q2q^2 for that example. The same question for the much smaller count |L|=q2|L|=q^2 has a longer history, coming from work on Kakeya sets. Wolff 1999 showed that a set L of q2q^2 lines in q3F_q^3, at most O(q)O(q) of which lie in any plane, has |P(L)|=Ω(q5/2)|P(L)|= (q^5/2), and Mockenhaupt & Tao 2004 showed that this is sharp when q is a square. Over prime fields the exponent improves: Ellenberg & Hablicsek 2016 prove that such an L has |P(L)|≥cq3|P(L)|≥ cq^3 for an absolute constant c>0c>0, once no plane contains more than q of its lines. Conjecture 1.4 assumes many more lines, Cq3Cq^3 rather than q2q^2, allows ω(q)ω(q) rather than O(q)O(q) of them in a plane, and asks for the constant 1−o(1)1-o(1) in place of c. We are aware of no counterexample in the literature respecting its plane hypothesis, and Tao 2025, who constructs a Nikodym set in qdF_q^d of size qd−((d−2)/log2+1+o(1))qd−1logq^d- ((d-2)/ 2+1+o(1) )q^d-1 q for each fixed d≥3d≥ 3 and each odd prime power q, still records Conjecture 1.4 as the route to the three-dimensional case of the Nikodym conjecture. The construction below is not new as a piece of finite geometry. The projective closure of the paraboloid is the quadric X2−νY2−ZW=0X^2-ν Y^2-ZW=0 of PG(3,q)PG(3,q), an elliptic quadric with q2+1q^2+1 points, and the splitting of the q+1q+1 tangent lines at each of its points into two halves of size (q+1)/2(q+1)/2, according to whether the quadratic form takes square or nonsquare values on the tangent line away from the quadric, is precisely the splitting used by Bruen & Drudge 1999 to build the first infinite family of Cameron–Liebler line classes of PG(3,q)PG(3,q), q odd, with parameter (q2+1)/2(q^2+1)/2. Gavrilyuk et al. 2018 write the splitting out explicitly as a partition of the tangent lines into two classes, either of which yields a Cameron–Liebler line class of that parameter when adjoined to the secant lines or to the external lines, and they derive a further such family from it. Cossidente & Pavese 2017 construct yet more families with the same parameter for odd q≥7q≥ 7. What is new here is only the observation that this classical half-tangent family, read in affine coordinates, is a counterexample to the line-union conjecture. Our counterexample is at the opposite extreme from the Hermitian family of Lund et al. 2018: its union misses a positive proportion of q3F_q^3 rather than a q−1/2q^-1/2 proportion, and it exists for every odd prime power, including the prime fields for which no Hermitian variety is available. Theorem C.10.1. Let q be an odd prime power. There is a set L of affine lines in q3F_q^3 such that |L|=q2(q+1)2,maxΠ|ℓ∈L:ℓ⊆Π|≤q+1,|P(L)|=q2(q+1)2,|L|= q^2(q+1)2, _ |\ ∈ L: \ |≤ q+1, |P(L)|= q^2(q+1)2, where the maximum is taken over all affine planes Π⊆q3 _q^3. Corollary C.10.2. Conjecture 1.4 of Lund et al. 2018 is false, and so is Conjecture 1.5 of Lund et al. 2018 for every fixed 0<ε<1/20< <1/2. Remark C.10.3. The construction requires q odd, since it selects half of the directions by a quadratic character and in characteristic 22 every element of qF_q is a square. This is no restriction for the purpose at hand: both conjectures are asymptotic statements as q→∞q→∞ over prime powers, so a counterexample along the odd prime powers refutes them. Remark C.10.4. Theorem C.10.1 closes the route to the three-dimensional Nikodym conjecture through Conjecture 1.4 of Lund et al. 2018, but says nothing about the Nikodym conjecture itself, which remains open. It shows only that the constant 11 in the conclusion of Conjecture 1.4 cannot be replaced by anything larger than 1/21/2; whether the conclusion survives with some absolute constant c>0c>0 in place of 11 is not addressed here, though Lund et al. 2018 already give c=0.38c=0.38 once C≥0.62C≥ 0.62. We now briefly summarize the construction. Fix a nonsquare ν∈q∗ν _q^* and let Q(x,y)=x2−νy2Q(x,y)=x^2-ν y^2, an anisotropic binary form, and slice q3F_q^3 into the q level sets of Q(x,y)−zQ(x,y)-z. The key point is that the tangent line to the paraboloid z=Q(x,y)z=Q(x,y) in a direction [u:v][u:v] meets only the level sets Q(x,y)−z=s\Q(x,y)-z=s\ with s=0s=0 or s in the square class of Q(u,v)Q(u,v), so keeping only the (q+1)/2(q+1)/2 directions with Q(u,v)Q(u,v) a square confines the whole family to the level sets indexed by 00 and by the nonzero squares. The plane bound comes from the fact that a plane meets the paraboloid in at most q+1q+1 points and, unless it is a tangent plane, determines the direction of a tangent line at each of them. C.10.2 The half-tangent family Fix an odd prime power q and a nonsquare ν∈q∗ν _q^*, and set Q(x,y)=x2−νy2.Q(x,y)=x^2-ν y^2. The binary form Q is anisotropic: if Q(x,y)=0Q(x,y)=0 with (x,y)≠(0,0)(x,y)≠(0,0), then y≠0y≠ 0 and ν=(x/y)2ν=(x/y)^2 would be a square. Let S=(a,b,Q(a,b)):a,b∈q⊆q3S=\(a,b,Q(a,b)):a,b _q\ _q^3 be the associated affine paraboloid, a set of q2q^2 points. Call a projective direction [u:v]∈PG(1,q)[u:v] (1,q) square if Q(u,v)Q(u,v) is a nonzero square in qF_q, and let D⊆PG(1,q)D (1,q) be the set of square directions. This is well defined: Q(u,v)≠0Q(u,v)≠ 0 for (u,v)≠(0,0)(u,v)≠(0,0) by anisotropy, and replacing (u,v)(u,v) by λ(u,v)λ(u,v) multiplies Q(u,v)Q(u,v) by the square λ2λ^2. Lemma C.10.5. We have |D|=(q+1)/2|D|=(q+1)/2. Proof. Since ν is a nonsquare we have q(ν)=q2F_q( ν)=F_q^2, and Q is the norm form of this extension: N(u+vν)=(u+vν)(u−vν)=u2−νv2=Q(u,v).N(u+v ν)=(u+v ν)(u-v ν)=u^2-ν v^2=Q(u,v). The norm map N:q2∗→q∗N:F_q^2^* _q^* is surjective with kernel of size q+1q+1, so every c∈q∗c _q^* has exactly q+1q+1 preimages. Hence exactly q−12(q+1) q-12(q+1) pairs (u,v)≠(0,0)(u,v)≠(0,0) have Q(u,v)Q(u,v) a nonzero square. Each projective direction accounts for exactly q−1q-1 such pairs, so |D|=(q+1)/2|D|=(q+1)/2. ∎ For (a,b)∈q2(a,b) _q^2 and [u:v]∈PG(1,q)[u:v] (1,q) set ℓa,b,[u:v]=(a+tu,b+tv,Q(a,b)+2t(au−νbv)):t∈q. _a,b,[u:v]= \ (a+tu,\ b+tv,\ Q(a,b)+2t(au-ν bv) ):t _q \. Replacing (u,v)(u,v) by λ(u,v)λ(u,v) only reparametrizes this set, so it depends on [u:v][u:v] alone, and it is a line because (u,v)≠(0,0)(u,v)≠(0,0). Expanding Q gives the identity Q(a+tu,b+tv)−(Q(a,b)+2t(au−νbv))=t2Q(u,v),Q(a+tu,\,b+tv)- (Q(a,b)+2t(au-ν bv) )=t^2Q(u,v), (29) so along ℓa,b,[u:v] _a,b,[u:v] the function Q(x,y)−zQ(x,y)-z equals t2Q(u,v)t^2Q(u,v) and vanishes to order two at t=0t=0. In other words, ℓa,b,[u:v] _a,b,[u:v] is the tangent line to S at (a,b,Q(a,b))(a,b,Q(a,b)) in the direction [u:v][u:v]. Define the half-tangent family L=ℓa,b,[u:v]:(a,b)∈q2,[u:v]∈D.L= \ _a,b,[u:v]:(a,b) _q^2,\ [u:v]∈ D \. Lemma C.10.6. Every ℓ∈L ∈ L meets S exactly in its point of tangency, and |L|=q2(q+1)/2|L|=q^2(q+1)/2. Proof. By equation 29, the point of ℓa,b,[u:v] _a,b,[u:v] with parameter t lies on S if and only if t2Q(u,v)=0t^2Q(u,v)=0. Since Q(u,v)≠0Q(u,v)≠ 0, this forces t=0t=0. So ℓ∩S ∩ S is the single point (a,b,Q(a,b))(a,b,Q(a,b)), which is therefore determined by ℓ , as is the direction [u:v][u:v]. Hence (a,b,[u:v])↦ℓa,b,[u:v](a,b,[u:v]) _a,b,[u:v] is injective on q2×DF_q^2× D, and Lemma C.10.5 gives |L|=q2⋅q+12|L|=q^2· q+12. ∎ Figure 11 shows the selected directions at a point of S, and the resulting union, in the case q=7q=7. p[1:0][1:0][1:1][1:1][1:2][1:2][1:3][1:3][1:4][1:4][1:5][1:5][1:6][1:6][0:1][0:1]s=0s=0s=1s=1s=2s=2s=3s=3s=4s=4s=5s=5s=6s=6 Figure 11: The two counts behind Theorem C.10.1, drawn for q=7q=7 and ν=3ν=3. On the left are the eight tangent lines to S at a point p, all lying in the tangent plane at p; the four solid ones are those whose direction [u:v][u:v] has Q(u,v)=u2−3v2Q(u,v)=u^2-3v^2 a nonzero square, and these are exactly the lines of L through p. On the right are the seven points of q3F_q^3 lying over a fixed (x,y)(x,y), labeled by the value s=Q(x,y)−zs=Q(x,y)-z; the four solid ones are those belonging to P(L)P(L), namely those with s∈0,1,2,4s∈\0,1,2,4\. C.10.3 Lines contained in a plane Lemma C.10.7. Every affine plane Π⊆q3 _q^3 contains at most q+1q+1 lines of L. Proof. Write Π=(x,y,z)∈q3:c1x+c2y+c3z=c0,(c1,c2,c3)≠(0,0,0). =\(x,y,z) _q^3:c_1x+c_2y+c_3z=c_0\, (c_1,c_2,c_3)≠(0,0,0). If ℓa,b,[u:v]⊆Π _a,b,[u:v] , then its point of tangency lies in Π∩S ∩ S and its direction vector (u,v,2(au−νbv))(u,v,2(au-ν bv)) lies in the direction plane of Π , the latter condition reading (c1+2c3a)u+(c2−2νc3b)v=0.(c_1+2c_3a)u+(c_2-2ν c_3b)v=0. (30) The size of Π∩S ∩ S. If c3=0c_3=0, then Π∩S ∩ S is parametrized by the q solutions (x,y)(x,y) of c1x+c2y=c0c_1x+c_2y=c_0, so |Π∩S|=q| ∩ S|=q. If c3≠0c_3≠ 0, then substituting z=Q(x,y)z=Q(x,y) into the equation of Π and completing the square gives Q(x+c12c3,y−c22νc3)=c0c3+c124c32−c224νc32.Q (x+ c_12c_3,\ y- c_22ν c_3 )= c_0c_3+ c_1^24c_3^2- c_2^24ν c_3^2. (31) If the right-hand side of equation 31 is zero, then anisotropy of Q gives exactly one solution, and otherwise the norm count in the proof of Lemma C.10.5 gives exactly q+1q+1 solutions. Hence |Π∩S|∈1,q,q+1| ∩ S|∈\1,q,q+1\, and in particular |Π∩S|≤q+1| ∩ S|≤ q+1. Tangent planes. Since Q(x,y)−zQ(x,y)-z has gradient (2a,−2νb,−1)(2a,-2ν b,-1) at (a,b,Q(a,b))(a,b,Q(a,b)), the tangent plane to S at that point is Ta,b:2ax−2νby−z=Q(a,b).T_a,b: 2ax-2ν by-z=Q(a,b). We claim that, for (a,b,Q(a,b))∈Π∩S(a,b,Q(a,b))∈ ∩ S, the two coefficients in equation 30 both vanish if and only if Π=Ta,b =T_a,b. Indeed, they vanish exactly when c1=−2c3ac_1=-2c_3a and c2=2νc3bc_2=2ν c_3b, which forces c3≠0c_3≠ 0; dividing the equation of Π by −c3-c_3 then turns it into 2ax−2νby−z=−c0/c32ax-2ν by-z=-c_0/c_3, and evaluating at (a,b,Q(a,b))∈Π(a,b,Q(a,b))∈ gives −c0/c3=Q(a,b)-c_0/c_3=Q(a,b), so Π=Ta,b =T_a,b. Conversely, if Π=Ta,b =T_a,b then (c1,c2,c3)=λ(2a,−2νb,−1)(c_1,c_2,c_3)=λ(2a,-2ν b,-1) for some λ∈q∗λ _q^*, whence c1=−2c3ac_1=-2c_3a and c2=2νc3bc_2=2ν c_3b. Comparing with equation 31, whose right-hand side vanishes exactly when the unique point of Π∩S ∩ S has (a,b)=(−c1/(2c3),c2/(2νc3))(a,b)=(-c_1/(2c_3),\,c_2/(2ν c_3)), we conclude that Π is a tangent plane of S if and only if |Π∩S|=1| ∩ S|=1. The two cases. Suppose first that Π is not a tangent plane of S. Then for every (a,b,Q(a,b))∈Π∩S(a,b,Q(a,b))∈ ∩ S the two coefficients in equation 30 are not both zero, so at most one [u:v]∈PG(1,q)[u:v] (1,q) satisfies equation 30, and therefore at most one line of L tangent at that point is contained in Π . Since every line of L contained in Π is tangent at some point of Π∩S ∩ S, the plane Π contains at most |Π∩S|≤q+1| ∩ S|≤ q+1 lines of L. Suppose instead that Π=Tp =T_p is the tangent plane at a point p∈Sp∈ S. Then Π∩S=p ∩ S=\p\, so every line of L inside Π is tangent at p, and there are exactly |D|=(q+1)/2|D|=(q+1)/2 of these. This exhausts all possibilities, and in every case Π contains at most q+1q+1 lines of L. ∎ C.10.4 The union Lemma C.10.8. We have P(L)=(x,y,z)∈q3:Q(x,y)−z∈0∪(q∗)2,P(L)= \(x,y,z) _q^3:\ Q(x,y)-z∈\0\∪(F_q^*)^2 \, and consequently |P(L)|=q2(q+1)/2|P(L)|=q^2(q+1)/2. Proof. Write s=Q(x,y)−zs=Q(x,y)-z for the value of the defining function at a point (x,y,z)(x,y,z). By equation 29, at the point of ℓa,b,[u:v] _a,b,[u:v] with parameter t we have s=t2Q(u,v)s=t^2Q(u,v). For [u:v]∈D[u:v]∈ D the value Q(u,v)Q(u,v) is a nonzero square, so s∈0∪(q∗)2s∈\0\∪(F_q^*)^2, which gives one inclusion. Conversely, suppose s=Q(x,y)−z∈0∪(q∗)2s=Q(x,y)-z∈\0\∪(F_q^*)^2, and fix any [u:v]∈D[u:v]∈ D, which exists by Lemma C.10.5. If s=0s=0, then (x,y,z)∈S(x,y,z)∈ S and (x,y,z)(x,y,z) is the point of ℓx,y,[u:v] _x,y,[u:v] with t=0t=0. If s≠0s≠ 0, then s/Q(u,v)s/Q(u,v) is a nonzero square, say s/Q(u,v)=λ2s/Q(u,v)=λ^2 with λ∈q∗λ _q^*, and we set a=x−λu,b=y−λv.a=x-λ u, b=y-λ v. At parameter t=λt=λ the line ℓa,b,[u:v] _a,b,[u:v] has first two coordinates (a+λu,b+λv)=(x,y)(a+λ u,b+λ v)=(x,y), and by equation 29 its third coordinate is Q(a,b)+2λ(au−νbv)=Q(x,y)−λ2Q(u,v)=Q(x,y)−s=z.Q(a,b)+2λ(au-ν bv)=Q(x,y)-λ^2Q(u,v)=Q(x,y)-s=z. Hence (x,y,z)∈ℓa,b,[u:v]⊆P(L)(x,y,z)∈ _a,b,[u:v] P(L), which gives the other inclusion. Finally, for each fixed (x,y)∈q2(x,y) _q^2 the map z↦Q(x,y)−z Q(x,y)-z is a bijection of qF_q, and |0∪(q∗)2|=1+q−12=q+12. |\0\∪(F_q^*)^2 |=1+ q-12= q+12. Therefore |P(L)|=q2⋅q+12|P(L)|=q^2· q+12, as desired. ∎ C.10.5 Proof of the main theorem Proof of Theorem C.10.1. Let L be the half-tangent family. Lemma C.10.6 gives |L|=q2(q+1)/2|L|=q^2(q+1)/2, Lemma C.10.7 gives maxΠ|ℓ∈L:ℓ⊆Π|≤q+1 _ |\ ∈ L: \|≤ q+1, and Lemma C.10.8 gives |P(L)|=q2(q+1)/2|P(L)|=q^2(q+1)/2. ∎ Proof of Corollary C.10.2. Let q range over the odd prime powers and let L=LqL=L_q be the family of Theorem C.10.1, so that |Lq|=q2(q+1)2≥q32,|P(Lq)|=q2(q+1)2=(12+o(1))q3.|L_q|= q^2(q+1)2≥ q^32, |P(L_q)|= q^2(q+1)2= ( 12+o(1) )q^3. For Conjecture 1.4, take C=1/2C=1/2. Given any function α with α(q)/q→∞α(q)/q→∞ we have α(q)>q+1α(q)>q+1 for all large q, so by Theorem C.10.1 no plane contains α(q)α(q) lines of LqL_q. The hypotheses therefore hold along the odd prime powers, while the conclusion |P(Lq)|≥(1−o(1))q3|P(L_q)|≥(1-o(1))q^3 fails. For Conjecture 1.5, fix 0<ε<1/20< <1/2. Then |Lq|≥q3/2≥q5/2+ε|L_q|≥ q^3/2≥ q^5/2+ as soon as q1/2−ε≥2q^1/2- ≥ 2, and q+1≤12q3/2q+1≤ 12q^3/2 for every q≥7q≥ 7, so the hypotheses hold for all sufficiently large odd q. But q3−|P(Lq)|=q32−q22,q^3-|P(L_q)|= q^32- q^22, which is not O(q5/2)O(q^5/2), so the conclusion fails. ∎ References. Bruen & Drudge (1999) Aiden A Bruen and Keldon Drudge. The construction of Cameron–Liebler line classes in PG(3,q)PG(3,q). Finite Fields and Their Applications, 5(1):35–45, 1999. Cossidente & Pavese (2017) Antonio Cossidente and Francesco Pavese. New Cameron–Liebler line classes with parameter q2+12 q^2+12. arXiv preprint arXiv:1707.01878, 2017. Ellenberg & Hablicsek (2016) Jordan S Ellenberg and Marton Hablicsek. An incidence conjecture of bourgain over fields of positive characteristic. In Forum of Mathematics, Sigma, volume 4, p. e23. Cambridge University Press, 2016. Gavrilyuk et al. (2018) Alexander L Gavrilyuk, Ilia Matkin, and Tim Penttila. Derivation of cameron–liebler line classes. Designs, Codes and Cryptography, 86(1):231–236, 2018. Lund et al. (2018) Ben Lund, Shubhangi Saraf, and Charles Wolf. Finite field kakeya and nikodym sets in three dimensions. SIAM Journal on Discrete Mathematics, 32(4):2836–2849, 2018. Mockenhaupt & Tao (2004) Gerd Mockenhaupt and Terence Tao. Restriction and Kakeya phenomena for finite fields. Duke Mathematical Journal, 121(1):35–74, 2004. Tao (2025) Terence Tao. New nikodym set constructions over finite fields. arXiv preprint arXiv:2511.07721, 2025. Wolff (1999) Thomas Wolff. Recent work connected with the kakeya problem. Prospects in mathematics (Princeton, NJ, 1996), 2:129–162, 1999. C.11 High-girth graphs attaining χcs(G)=2χ(G)χ^s_c(G)=2χ(G) For every graph G the signed circular chromatic number satisfies χc(G)≤χcs(G)≤2χc(G) _c(G)≤χ^s_c(G)≤ 2 _c(G), and Naserasr et al. 2020 showed that the upper bound is approached by k-chromatic graphs of arbitrarily large girth. Whether it is reached by a finite such graph was left open. We show that it is: for all integers k,g≥2k,g≥ 2 there is a finite simple graph of chromatic number k and girth at least g whose signed circular chromatic number is exactly 2k2k. The witness is a sparse random k-partite signed graph with its short cycles deleted, and the point of the proof is a union bound that rules out every admissible ratio p/q<2kp/q<2k at once. C.11.1 Introduction A signed graph (G,σ)(G,σ) is a graph G together with a signature σ:E(G)→+,−σ:E(G)→\+,-\. Following Naserasr et al. 2020, for an even integer p and an integer q with 1≤q≤p/21≤ q≤ p/2, a (p,q)(p,q)-coloring of (G,σ)(G,σ) is a map f:V(G)→ℤpf:V(G) _p such that dp(f(u),f(v))≥qfor every positive edge uv,d_p (f(u),f(v) )≥ q every positive edge uv, dp(f(u),f(v)+p2)≥qfor every negative edge uv,d_p (f(u),f(v)+ p2 )≥ q every negative edge uv, where dp(a,b)=min|a−b|,p−|a−b|d_p(a,b)= \|a-b|,\,p-|a-b|\ is the distance in the cycle ℤpZ_p. The circular chromatic number of (G,σ)(G,σ) is χc(G,σ)=infp/q:(G,σ) has a (p,q)-coloring _c(G,σ)= \p/q:(G,σ) has a (p,q)-coloring\, and the signed circular chromatic number of a graph G is χcs(G)=maxχc(G,σ):σ a signature of G.χ^s_c(G)= \ _c(G,σ):σ a signature of G \. For the all-positive signature one recovers the ordinary circular chromatic number surveyed by Zhu 2001, so χc(G)≤χcs(G) _c(G)≤χ^s_c(G), and Naserasr et al. 2020 give the matching upper bound χcs(G)≤2χc(G)χ^s_c(G)≤ 2 _c(G). Switching at a vertex set A, that is, reversing the signs of the edges of the cut (A,V(G)∖A)(A,V(G) A), in the sense of Zaslavsky 1982b, does not change χc(G,σ) _c(G,σ), since it amounts to replacing f(v)f(v) by f(v)+p/2f(v)+p/2 on A; quantifying over all maps f:V(G)→ℤpf:V(G) _p therefore already accounts for switching. Throughout, the girth of a signed graph is the girth of its underlying graph, and all graphs are simple, so girth at least 33 is automatic and digons do not arise. Naserasr, Wang and Zhu proved that the bound χcs(G)≤2χc(G)χ^s_c(G)≤ 2 _c(G) remains tight when the girth is prescribed: for all integers k,g≥2k,g≥ 2 and every ε>0 >0 there is a graph G of girth at least g with χ(G)=kχ(G)=k and χcs(G)>2k−εχ^s_c(G)>2k- (Naserasr et al. 2020, Theorem 30). For each integer p they produce a graph on the vertex set of a bipartite augmented tree of Alon et al. 2016, with one new edge for each leaf joining two of that leaf’s ancestors, carrying a signature that admits no (2kp,p+1)(2kp,p+1)-coloring. The resulting lower bounds 2kp/(p+1)2kp/(p+1) increase to 2k2k but never equal it. In the remark immediately following that proof they ask whether the supremum is attained: It is not known whether there is a finite k-chromatic graph of girth at least g and with χcs(G)=2kχ^s_c(G)=2k. We answer this affirmatively, for every pair k,g≥2k,g≥ 2, by a random construction. Attainment questions of this shape go both ways in this subject. On the one hand, Naserasr et al. 2020 show that the supremum of χc(G,σ) _c(G,σ) over signed d-degenerate simple graphs equals 2⌊d/2⌋+22 d/2 +2, and that it is attained by (Kd+1,+)(K_d+1,+) for odd d and by an explicit signed graph Ωd _d for even d≥4d≥ 4; for d=2d=2 they produce only a sequence of signed graphs whose circular chromatic numbers tend to 44. Kardoš et al. 2023 then showed that in the case d=2d=2 the supremum is genuinely never attained, by proving that every signed 22-degenerate simple graph on n vertices has circular chromatic number at most 4−2/⌊(n+1)/2⌋4-2/ (n+1)/2 , and that this bound is tight for every n≥2n≥ 2. On the other hand Naserasr et al. 2020 attain the supremum 10/310/3 for signed series-parallel simple graphs with an explicit signed outerplanar graph, and Pan & Zhu 2022 went on to show that every rational in [2,10/3][2,10/3] occurs; Zhu & Zhu 2023 carry the same analysis out for signed series-parallel graphs in which every cycle with an odd number of positive edges is long, a signed analogue of odd girth rather than the girth of the underlying graph. The girth family of Naserasr et al. 2020 sat on neither side of this divide. Coloring of signed graphs goes back to Zaslavsky 1982a, whose 00-free 2k2k-colorings are exactly the circular 2k2k-colorings (Naserasr et al. 2020), and to Máčajová et al. 2014, who proposed a chromatic number for signed graphs and conjectured that every signed planar simple graph is 44-colorable, equivalently that χcs(G)≤4χ^s_c(G)≤ 4 for every planar G (Naserasr et al. 2020); Kardoš & Narboni 2021 refuted this, and Naserasr et al. 2020 exhibit a signed planar simple graph with χc=4+23 _c=4+ 23. A different circular refinement of signed graph coloring was introduced earlier by Kang & Steffen 2018; the two notions are compared by Naserasr et al. 2020, and the area as a whole is surveyed in Wang 2022. We are not aware of any work that settles the attainment question above for every pair k,gk,g. Our construction adapts the deletion argument of Erdös 1959 to a random k-partite signed graph. C.11.2 Statement and small cases One half of the problem is immediate. Lemma C.11.1. If χ(G)≤kχ(G)≤ k then every signature σ of G admits a (2k,1)(2k,1)-coloring, and hence χcs(G)≤2kχ^s_c(G)≤ 2k. Proof. Let V1,…,VkV_1,…,V_k be the color classes of a proper k-coloring of G and put f(v)=i−1∈ℤ2kf(v)=i-1 _2k for v∈Viv∈ V_i. Let uv∈E(G)uv∈ E(G). Then f(u)≠f(v)f(u)≠ f(v), so d2k(f(u),f(v))≥1d_2k(f(u),f(v))≥ 1; and f(v)+k∈k,…,2k−1f(v)+k∈\k,…,2k-1\ is distinct from f(u)∈0,…,k−1f(u)∈\0,…,k-1\, so d2k(f(u),f(v)+k)≥1d_2k(f(u),f(v)+k)≥ 1. Both edge conditions hold regardless of σ, so χc(G,σ)≤2k _c(G,σ)≤ 2k for every σ. ∎ For small k and g the remaining half can already be settled by a finite search. Example C.11.2. Take k=2k=2 and g=4g=4, and let K3,4K_3,4 have parts u1,u2,u3\u_1,u_2,u_3\ and w1,w2,w3,w4\w_1,w_2,w_3,w_4\. The signature whose negative edges are exactly u2w2u_2w_2, u2w4u_2w_4, u3w2u_3w_2 and u3w3u_3w_3 has circular chromatic number 44, so χcs(K3,4)=4=2χ(K3,4)χ^s_c(K_3,4)=4=2χ(K_3,4). Such a value is certified by an exhaustive check over the finitely many pairs (p,q)(p,q) left admissible by Naserasr et al. 2020, which states that χc(G,σ)=p/q _c(G,σ)=p/q for some even p≤2|V(G)|p≤ 2|V(G)| and that the infimum in its definition is a minimum. No graph on fewer than seven vertices has χ=2χ=2 and χcs=4χ^s_c=4. Indeed, if H⊆GH G then every signature of H extends to G and every (p,q)(p,q)-coloring of the extension restricts to H, so χcs(H)≤χcs(G)χ^s_c(H)≤χ^s_c(G); every bipartite graph on at most six vertices is a subgraph of K3,3K_3,3, of K2,4K_2,4 or of K1,5K_1,5; and χcs(K3,3)=3χ^s_c(K_3,3)=3, χcs(K2,4)=8/3χ^s_c(K_2,4)=8/3 and χcs(K1,5)=2χ^s_c(K_1,5)=2. The value χcs(K3,4)=4χ^s_c(K_3,4)=4 is also due to Gujgiczer et al. 2023, whose signed graph BQ^(2,3) BQ(2,3) is K3,4K_3,4 with a maximum matching positive and the remaining nine edges negative; naming that matching u1w1u_1w_1, u2w3u_2w_3, u3w4u_3w_4 and switching at u1,w1\u_1,w_1\ turns it into the signature above. At k=3k=3 and g=3g=3 the same happens: if K3,3,4K_3,3,4 has parts x1,x2,x3\x_1,x_2,x_3\, y1,y2,y3\y_1,y_2,y_3\ and z1,z2,z3,z4\z_1,z_2,z_3,z_4\, then the signature whose negative edges are exactly x1y1,x1y3,x3y3,x1z4,x2z1,x2z4,y1z2,y1z4,y2z2x_1y_1,\ x_1y_3,\ x_3y_3,\ x_1z_4,\ x_2z_1,\ x_2z_4,\ y_1z_2,\ y_1z_4,\ y_2z_2 has circular chromatic number 66, so χcs(K3,3,4)=6=2χ(K3,3,4)χ^s_c(K_3,3,4)=6=2χ(K_3,3,4). Theorem C.11.3. For all integers k,g≥2k,g≥ 2 there is a finite simple graph G with χ(G)=kχ(G)=k, girth at least g, and χcs(G)=2kχ^s_c(G)=2k. Remark C.11.4. The proof is a probabilistic existence argument and exhibits no explicit graph. All three estimates it uses are effective, so a bound on the least admissible |V(G)||V(G)| in terms of k and g could be extracted. Beyond the witnesses of Example C.11.2, which have girth 33 and 44, we know of no explicit witness of girth at least 55. The same remark of Naserasr et al. 2020 asks a second question, namely whether for every rational p/qp/q, every integer g and every ε>0 >0 there is a graph G of girth at least g with χc(G)≤p/q _c(G)≤ p/q and χcs(G)>2p/q−εχ^s_c(G)>2p/q- . The construction below is tied to an integer k through the k-partition and through Lemma C.11.1, and does not address that question. By Lemma C.11.1 the whole content of Theorem C.11.3 is the lower bound: one must exhibit a single signature σ on a k-chromatic graph of girth at least g with χc(G,σ)≥2k _c(G,σ)≥ 2k, that is, with no (p,q)(p,q)-coloring for any admissible (p,q)(p,q) with p/q<2kp/q<2k. Two features of the problem make this delicate. First, a k-critical graph satisfies χcs(G)≤2k−2χ^s_c(G)≤ 2k-2 (Naserasr et al. 2020), so a witness must be far from critical; the construction below in fact produces graphs in which every single edge can be deleted without lowering the chromatic number. Second, by Naserasr et al. 2020 the value χc(G,σ) _c(G,σ) is a ratio p/qp/q with p even and p≤2|V(G)|p≤ 2|V(G)|, so the number of ratios below 2k2k that have to be excluded grows with the graph. Hence an argument that excludes one fixed target ratio per construction cannot suffice, and the union bound has to range over all admissible (p,q)(p,q) at once. The proof splits into two independent pieces. The first is a deterministic statement about colors: if p/q<2kp/q<2k then among any k colors in ℤpZ_p there are two that one of the two signs forbids, so every k-tuple of colors carries a blocking pair. The second is the usual Erdős deletion argument, applied to a k-partite random signed graph sparse enough that short cycles are rare but dense enough that the failure probability for a fixed coloring beats the number of colorings. C.11.3 Neutral color pairs Fix an even integer p and an integer q with 1≤q≤p/21≤ q≤ p/2. Call an unordered pair a,b⊆ℤp\a,b\ _p neutral if it obstructs neither sign, that is, if dp(a,b)≥qd_p(a,b)≥ q and dp(a,b+p2)≥qd_p(a,b+ p2)≥ q. Since dp(a,b+p2)=p2−dp(a,b)d_p(a,b+ p2)= p2-d_p(a,b), neutrality says exactly that q≤dp(a,b)≤p2−q.q\ ≤\ d_p(a,b)\ ≤\ p2-q. (32) A pair that is not neutral is blocking: at least one of the two signs placed on it violates the (p,q)(p,q)-condition. Note that a,a\a,a\ is blocking, because dp(a,a)=0<qd_p(a,a)=0<q. aabbAaA_aAaA_aAbA_bAbA_b Figure 12: The two arcs making up AaA_a and the two arcs making up AbA_b, drawn on the cycle ℤpZ_p. Each arc has length q, so |Aa|=|Ab|=2q|A_a|=|A_b|=2q; as in the proof of Lemma C.11.5, the pair a,b\a,b\ is neutral precisely when the four arcs are pairwise disjoint, which is the situation drawn. Lemma C.11.5. The graph on vertex set ℤpZ_p whose edges are the neutral pairs has clique number at most ⌊p/(2q)⌋ p/(2q) . In particular, if p/q<2kp/q<2k then it contains no clique of size k. Proof. For a∈ℤpa _p let Aa=[a,a+q)∪[a+p2,a+p2+q)A_a=[a,a+q)∪[a+ p2,a+ p2+q), a union of two arcs of ℤpZ_p of length q each, as in Figure 12. The two arcs are disjoint because q≤p/2q≤ p/2, so |Aa|=2q|A_a|=2q. Suppose a,b\a,b\ is neutral. The arcs [a,a+q)[a,a+q) and [b,b+q)[b,b+q) meet only if dp(a,b)<qd_p(a,b)<q, which equation 32 forbids. The arcs [a,a+q)[a,a+q) and [b+p2,b+p2+q)[b+ p2,b+ p2+q) meet only if dp(a,b+p2)<qd_p(a,b+ p2)<q, likewise forbidden, and the two remaining pairs of arcs give back these same two conditions. Hence Aa∩Ab=∅A_a∩ A_b= . If a1,…,ata_1,…,a_t is a neutral clique, the sets Aa1,…,AatA_a_1,…,A_a_t are therefore pairwise disjoint subsets of ℤpZ_p, so 2tq≤p2tq≤ p and t≤⌊p/(2q)⌋t≤ p/(2q) . Finally p/q<2kp/q<2k gives ⌊p/(2q)⌋≤k−1 p/(2q) ≤ k-1. ∎ Lemma C.11.5 is the only place where the hypothesis p/q<2kp/q<2k is used, and it is what forces every k-tuple of colors to contain a blocking pair. C.11.4 The random construction Fix k≥2k≥ 2 and g≥2g≥ 2, and fix a real number α with 0<α<1,α(g−2)<1.0<α<1, α(g-2)<1. For a large integer m set n=kmn=km and ρ=m−1+α∈(0,1]ρ=m^-1+α∈(0,1]. Let V=V1∪⋯∪VkV=V_1∪…∪ V_k with |Vi|=m|V_i|=m, so |V|=n|V|=n, and call a pair of vertices lying in distinct parts a cross pair. Independently for each cross pair u,v\u,v\, place a positive edge with probability ρ2,a negative edge with probability ρ2, positive edge with probability ρ2, negative edge with probability ρ2, no edge with probability 1−ρ. edge with probability 1-ρ. No edge is placed inside a part. This yields a random signed simple graph (G0,σ0)(G_0, _0) on V whose underlying graph is k-partite. Lemma C.11.6. Let p be even, let 1≤q≤p/21≤ q≤ p/2 with p/q<2kp/q<2k, and let f:V→ℤpf:V _p be arbitrary. Then at least m2m^2 cross pairs u,v\u,v\ have f(u),f(v)\f(u),f(v)\ blocking. Likewise, for every map c:V→[k−1]c:V→[k-1] at least m2m^2 cross pairs are monochromatic under c. Proof. Call a tuple (v1,…,vk)∈V1×⋯×Vk(v_1,…,v_k)∈ V_1×…× V_k a transversal; there are mkm^k of them. If all (k2) k2 pairs of a transversal were neutral, then the colors f(v1),…,f(vk)f(v_1),…,f(v_k) would be pairwise distinct, since a repeated color gives a blocking pair, and they would form a neutral clique of size k, contradicting Lemma C.11.5. So every transversal contains a blocking cross pair. Each cross pair lies in exactly mk−2m^k-2 transversals, obtained by choosing one vertex from each of the remaining k−2k-2 parts. Hence the number of blocking cross pairs is at least mk/mk−2=m2m^k/m^k-2=m^2. The second statement is the same count, with the pigeonhole principle in place of Lemma C.11.5. ∎ C.11.5 Proof of the main theorem Proof of Theorem C.11.3. We consider three events for (G0,σ0)(G_0, _0). No cheap circular coloring. Fix an even p and an integer 1≤q≤p/21≤ q≤ p/2 with p/q<2kp/q<2k, and fix f:V→ℤpf:V _p. Let XfX_f be the number of edges of (G0,σ0)(G_0, _0) violating f. By Lemma C.11.6 we may fix a set of exactly m2m^2 blocking cross pairs; for each of them one of the two signs violates f, and that particular signed edge is present with probability ρ/2ρ/2, independently over pairs. Hence XfX_f stochastically dominates a Bin(m2,ρ/2)Bin(m^2,ρ/2) variable, whose mean is 12m1+α 12m^1+α, and the Chernoff bound gives ℙ[Xf<14m1+α]≤exp(−116m1+α).P [X_f< 14m^1+α ]\ ≤\ (- 116m^1+α ). By Naserasr et al. 2020 the circular chromatic number of any signed graph on n vertices equals p/qp/q for some even p≤2np≤ 2n, and this applies to every spanning subgraph of G0G_0, so it suffices to range over even p≤2np≤ 2n. The number of triples (p,q,f)(p,q,f) with p even, p≤2np≤ 2n, 1≤q≤p/21≤ q≤ p/2 and f:V→ℤpf:V _p is at most n⋅n⋅(2n)n=exp(O(mlogm))n· n·(2n)^n= (O(m m)), and m1+α≫mlogm^1+α m m, so with probability 1−o(1)1-o(1) every such triple with p/q<2kp/q<2k satisfies Xf≥14m1+αX_f≥ 14m^1+α. No cheap (k−1)(k-1)-coloring. For a map c:V→[k−1]c:V→[k-1] let YcY_c be the number of c-monochromatic edges of G0G_0. By Lemma C.11.6 and the same argument, YcY_c stochastically dominates Bin(m2,ρ)Bin(m^2,ρ), whose mean is m1+αm^1+α, so ℙ[Yc<14m1+α]≤exp(−18m1+α)P[Y_c< 14m^1+α]≤ (- 18m^1+α). There are (k−1)n=exp(O(m))(k-1)^n= (O(m)) such maps, so with probability 1−o(1)1-o(1) every c has Yc≥14m1+αY_c≥ 14m^1+α. Few short cycles. Let Z be the number of cycles of G0G_0 of length less than g. Since a cycle of length ℓ is present only if all ℓ of its pairs receive edges, [Z]≤∑ℓ=3g−1nℓρℓ2ℓ≤∑ℓ=3g−1(kmα)ℓ=O(mα(g−1)),E[Z]\ ≤\ _ =3^g-1 n ρ 2 \ ≤\ _ =3^g-1(km^α) \ =\ O (m^α(g-1) ), where the implicit constant depends on k and g only. Now α(g−1)<1+α(g-1)<1+α precisely because α(g−2)<1α(g-2)<1, so [Z]=o(m1+α)E[Z]=o(m^1+α), and Markov’s inequality gives Z<18m1+αZ< 18m^1+α with probability 1−o(1)1-o(1). For m large all three events hold simultaneously, and we fix such an outcome. Delete one edge from each cycle of G0G_0 of length less than g, obtaining a signed graph (G,σ)(G,σ) on the same vertex set V with fewer than 18m1+α 18m^1+α edges removed. A cycle of G of length less than g would be a cycle of G0G_0 of length less than g all of whose edges survived the deletion, and there is none, so G has girth at least g; moreover G is simple and k-partite. Let p be even with p≤2np≤ 2n, let 1≤q≤p/21≤ q≤ p/2 satisfy p/q<2kp/q<2k, and let f:V→ℤpf:V _p be arbitrary. Then f still violates at least 14m1+α−18m1+α>0 14m^1+α- 18m^1+α>0 edges of (G,σ)(G,σ), so (G,σ)(G,σ) has no (p,q)(p,q)-coloring for any such pair. Since |V(G)|=n|V(G)|=n, Naserasr et al. 2020 gives χc(G,σ)=p0/q0 _c(G,σ)=p_0/q_0 for a pair (p0,q0)(p_0,q_0) with p0p_0 even, p0≤2np_0≤ 2n and 1≤q0≤p0/21≤ q_0≤ p_0/2 for which (G,σ)(G,σ) has a (p0,q0)(p_0,q_0)-coloring, and the previous sentence forces p0/q0≥2kp_0/q_0≥ 2k, so χc(G,σ)≥2k _c(G,σ)≥ 2k. Similarly every c:V→[k−1]c:V→[k-1] leaves a monochromatic edge, so χ(G)≥kχ(G)≥ k, while χ(G)≤kχ(G)≤ k because G is k-partite. Finally Lemma C.11.1 gives χcs(G)≤2kχ^s_c(G)≤ 2k, whence 2k≤χc(G,σ)≤χcs(G)≤ 2k.∎2k\ ≤\ _c(G,σ)\ ≤\ χ^s_c(G)\ ≤\ 2k. Remark C.11.7. The graph produced above is not merely non-critical. Indeed, every map c:V→[k−1]c:V→[k-1] leaves at least 18m1+α>1 18m^1+α>1 monochromatic edges of G, so χ(G−e)=kχ(G-e)=k for every edge e of G. References. Alon et al. (2016) Noga Alon, Alexandr Kostochka, Benjamin Reiniger, Douglas B West, and Xuding Zhu. Coloring, sparseness and girth. Israel Journal of Mathematics, 214(1):315–331, 2016. Erdös (1959) Paul Erdös. Graph theory and probability. Canadian Journal of Mathematics, 11:34–38, 1959. Gujgiczer et al. (2023) Anna Gujgiczer, Reza Naserasr, S Taruni, et al. Winding number and circular 4-coloring of signed graphs. arXiv preprint arXiv:2307.04652, 2023. Kang & Steffen (2018) Yingli Kang and Eckhard Steffen. Circular coloring of signed graphs. Journal of Graph Theory, 87(2):135–148, 2018. Kardoš & Narboni (2021) František Kardoš and Jonathan Narboni. On the 4-color theorem for signed graphs. European Journal of Combinatorics, 91:103215, 2021. Kardoš et al. (2023) František Kardoš, Jonathan Narboni, Reza Naserasr, and Zhouningxin Wang. Circular-coloring of some classes of signed graphs. SIAM Journal on Discrete Mathematics, 37(2):1198–1211, 2023. Máčajová et al. (2014) Edita Máčajová, André Raspaud, and Martin Škoviera. The chromatic number of a signed graph. arXiv preprint arXiv:1412.6349, 2014. Naserasr et al. (2020) Reza Naserasr, Zhouningxin Wang, and Xuding Zhu. Circular chromatic number of signed graphs. arXiv preprint arXiv:2010.07525, 2020. Pan & Zhu (2022) Zhishi Pan and Xuding Zhu. The circular chromatic numbers of signed series-parallel graphs. Discrete Mathematics, 345(3):112733, 2022. Wang (2022) Zhouningxin Wang. Circular coloring, circular flow, and homomorphism of signed graphs. PhD thesis, Université Paris Cité, 2022. Zaslavsky (1982a) Thomas Zaslavsky. Signed graph coloring. Discrete Mathematics, 39(2):215–228, 1982a. Zaslavsky (1982b) Thomas Zaslavsky. Signed graphs. Discrete Applied Mathematics, 4(1):47–74, 1982b. Zhu & Zhu (2023) Jialu Zhu and Xuding Zhu. The circular chromatic number of signed series–parallel graphs of given girth. Discrete Applied Mathematics, 341:82–92, 2023. Zhu (2001) Xuding Zhu. Circular chromatic number: a survey. Discrete mathematics, 229(1-3):371–410, 2001. C.12 Counting 4-critical linear triple systems Rödl and Siggers conjectured that the number of k-critical (r,l)(r,l)-systems on n vertices is exponential in nln^l. We disprove this for (k,r,l)=(4,3,2)(k,r,l)=(4,3,2): there is a constant c>0c>0 such that for infinitely many n there are at least exp(cn2logn) (cn^2 n) pairwise nonisomorphic 44-critical linear triple systems on n vertices. The construction lifts a dense 44-critical graph to a triple system along a proper edge coloring of that graph, in such a way that the edge coloring can be read off from any minimal non-33-colorable subsystem of the lift. The extra logn n in the exponent is exactly the entropy of the edge coloring, and it survives the passage from labeled systems to isomorphism classes. C.12.1 Introduction A hypergraph H is k-colorable if its vertices can be colored with k colors so that no edge is monochromatic, and k-chromatic if k is the least such number. Following Rödl & Siggers 2006, H is k-critical if it is k-chromatic, has no isolated vertices, and H−eH-e is (k−1)(k-1)-colorable for every edge e of H. An (r,l)(r,l)-system is an r-uniform hypergraph in which no l-set of vertices lies in more than one edge; a (3,2)(3,2)-system is a linear triple system. Write T(k,r,l,n)T(k,r,l,n) for the number of nonisomorphic k-critical (r,l)(r,l)-systems on n vertices. The first bound on this count is due to Abbott et al. 1980, who showed that for all k,r≥3k,r≥ 3 there is a constant b=b(k,r)>1b=b(k,r)>1 with T(k,r,2,n)>bnT(k,r,2,n)>b^\,n for all large n. Rödl & Siggers 2006 proved that for all k≥3k≥ 3 and r>l≥2r>l≥ 2 and all large n there is a k-critical (r,l)(r,l)-system on n vertices with at least c0nlc_0\,n^l edges, which is optimal up to the constant since an (r,l)(r,l)-system on n vertices has at most (nl)/(rl) nl / rl edges. Feeding that density into a construction indexed by the bipartitions of the edge set, they improved the Abbott–Liu–Toft bound to T(k,r,l,n)>αnlT(k,r,l,n)>α^\,n^l for all k≥3k≥ 3, r>l≥2r>l≥ 2 and all large n, with a constant α=α(k,r,l)>1α=α(k,r,l)>1 (Rödl & Siggers 2006, Theorem 6.1). Every system they produce has at least c′nlc n^l edges for a constant c′>0c >0, and Section 6 of Rödl & Siggers 2006 closes with a trivial upper bound and a conjecture: A trivial upper bound for the number of (r,l)(r,l)-systems with c′nlc n^l edges is ((nr)c′nl)<(nrc′nl)≈(nr−l)nl=O(dnllognr−l) nrc n^l< n^rc n^l≈(n^r-l)^n^l=O(d^\,n^l n^r-l). We conjecture that the actual number is in fact exponential in nln^l. We read “the actual number” as T(k,r,l,n)T(k,r,l,n), which the preceding theorem bounds from below by αnlα^\,n^l; the content of the conjecture is then the matching upper bound T(k,r,l,n)≤CnlT(k,r,l,n)≤ C^n^l for some constant C=C(k,r,l)C=C(k,r,l), that is, that the factor lognr−l n^r-l in the trivial bound is an artifact of the counting. For graphs the corresponding assertion is immediate, since there are only 2(n2)2 n2 graphs on n labeled vertices. Once r>lr>l, the trivial count of (r,l)(r,l)-systems with Θ(nl) (n^l) edges carries a logarithm in the exponent, and the conjecture asserts that criticality removes it. We show that criticality does not remove it, already for linear triple systems and k=4k=4. On the competing reading, in which “the actual number” counts all (r,l)(r,l)-systems on n vertices with Ω(nl) (n^l) edges, criticality plays no role and the logarithm is present for a direct counting reason, so it is the reading above that is at issue. Beyond the lower bound αnlα^\,n^l of Rödl & Siggers 2006 we are aware of no further work on the growth of T(k,r,l,n)T(k,r,l,n), and the problem is not recorded in the survey of color-critical hypergraphs of Kostochka 2006, which concerns critical systems with few edges rather than their number. Theorem C.12.1. There is a constant c>0c>0 and there are infinitely many integers n such that T(4,3,2,n)≥exp(cn2logn).T(4,3,2,n)≥ (c\,n^2 n ). In particular there is no constant C for which T(4,3,2,n)≤Cn2T(4,3,2,n)≤ C^n^2 for all n. We now summarize the construction. Fix a 44-critical graph G on m vertices with Ω(m2) (m^2) edges, which exists by a theorem of Toft, and fix a palette [t][t] with t=4mt=4m. Every proper edge coloring β:E(G)→[t]β E(G)→[t] is turned into a linear triple system JβJ_β on a vertex set that does not depend on β: take three copies v1,v2,v3v^1,v^2,v^3 of each vertex v of G together with one palette vertex wjw_j for each j∈[t]j∈[t], and place over each edge uvuv of G the three triples ui,vi,wβ(uv)\u^i,v^i,w_β(uv)\. Two forcing gadgets of Rödl & Siggers 2006 are then attached: one chains the palette vertices so that they all receive a common color, and one ties v1,v2,v3v^1,v^2,v^3 together so that they receive three distinct colors. A proper 33-coloring of JβJ_β would therefore select, for each v, the unique copy viv^i carrying the palette color, and v↦iv i would be a proper 33-coloring of G. The key point is that a minimal non-33-colorable subsystem KβK_β of JβJ_β must retain a triple over every edge of G, since G is 44-critical; that triple exhibits the value β(uv)β(uv), so β↦Kβ K_β is injective on labeled systems. There are exp(Ω(m2logm)) ( (m^2 m)) choices of β, while JβJ_β has only O(m)O(m) vertices, so passing to isomorphism classes costs a factor exp(O(mlogm)) (O(m m)) and the bound survives. C.12.2 The construction We use two known ingredients. The first is a theorem of Toft 1970: there is a constant a>0a>0 such that for every sufficiently large m there is a 44-critical graph on m vertices with at least am2a\,m^2 edges. As recorded by Rödl & Siggers 2006, one may take a=1/16a=1/16 here. The second is the pair of forcing gadgets of Rödl & Siggers 2006. We recall only the properties used below. Lemma C.12.2 (Forcing gadgets). There are fixed finite linear triple systems S and D with the following properties. 1. S has two designated vertices x,yx,y, and no edge of S contains both of them. In every proper 33-coloring of S the vertices x,yx,y receive the same color, and conversely every assignment of a common color to x and y extends to a proper 33-coloring of S. 2. D has three designated vertices u1,u2,u3u_1,u_2,u_3. In every proper 33-coloring of D these three vertices receive three distinct colors, and conversely every assignment of three distinct colors to u1,u2,u3u_1,u_2,u_3 extends to a proper 33-coloring of D. Proof. Take S=S(3,3)S=S(3,3) and D=D(3,3)D=D(3,3) from Rödl & Siggers 2006. Both are 33-chromatic (3,2)(3,2)-systems, so each has at least one proper 33-coloring, and the two forward implications are exactly the forcing properties for which those gadgets are built. That no edge of S contains both designated vertices is noted in the proof of that construction: S(3,3)S(3,3) is obtained by deleting edges from the system left when one edge through x and y is removed from a 44-critical linear triple system, and such a system exists by Abbott & Liu 1978. For the converse statements, fix a proper 33-coloring ψ0 _0 of S. Then ψ0(x)=ψ0(y) _0(x)= _0(y), and given any target color γ we may compose ψ0 _0 with a permutation of [3][3] carrying ψ0(x) _0(x) to γ. Likewise fix a proper 33-coloring ψ0 _0 of D; then ψ0(u1),ψ0(u2),ψ0(u3) _0(u_1), _0(u_2), _0(u_3) are distinct, and given distinct target colors γ1,γ2,γ3 _1, _2, _3 the assignment ψ0(ui)↦γi _0(u_i) _i is a well-defined permutation of [3][3], with which we compose ψ0 _0. ∎ Fix from now on an integer m large enough for Toft’s theorem, and let G be a 44-critical graph on m vertices with e(G)≥am2.e(G)≥ a\,m^2. Set t=4mt=4m, and let A be the set of proper edge colorings β:E(G)→[t]β E(G)→[t], that is, of maps assigning distinct colors to any two edges sharing an endpoint. Lemma C.12.3. ||≥(2m)e(G)|A|≥(2m)^e(G). Proof. Order E(G)E(G) arbitrarily and color greedily. When the edge uvuv is reached, the colors already used on edges meeting uvuv number at most (degu−1)+(degv−1)≤2m−4,( u-1)+( v-1)≤ 2m-4, since every degree is at most m−1m-1. As t=4mt=4m, at least 2m+42m+4 colors are available at every step. Distinct sequences of choices produce distinct colorings, since the edge order is fixed, so ||≥(2m)e(G)|A|≥(2m)^e(G). ∎ We now define the ambient vertex set. Let V∗=v1,v2,v3:v∈V(G)∪w1,…,wt∪W,V =\v^1,v^2,v^3:v∈ V(G)\∪\w_1,…,w_t\∪ W, where W consists of t−1t-1 disjoint sets of fresh internal vertices, one for a copy of S attached to each consecutive pair wj,wj+1w_j,w_j+1, together with m further disjoint sets of fresh internal vertices, one for a copy of D attached to each triple v1,v2,v3v^1,v^2,v^3. The set V∗V does not depend on β. For β∈β let JβJ_β be the hypergraph on V∗V whose edges are the following. For every uv∈E(G)uv∈ E(G) and every i∈[3]i∈[3] there is the lifted triple ui,vi,wβ(uv).\u^i,v^i,w_β(uv)\. (33) For every j∈[t−1]j∈[t-1] there are the edges of a copy of S whose designated vertices are identified with wjw_j and wj+1w_j+1 and whose remaining vertices are the corresponding fresh set in W. For every v∈V(G)v∈ V(G) there are the edges of a copy of D whose designated vertices are identified with v1,v2,v3v^1,v^2,v^3 and whose remaining vertices are the corresponding fresh set in W. Figure 13 shows the three pieces. u1u^1u2u^2u3u^3v1v^1v2v^2v3v^3wβ(uv)w_β(uv)the three triples over uvuvSSSSSSw1w_1w2w_2w3w_3wtw_tone common color on the paletteDDv1v^1v2v^2v3v^3three distinct colors on the copies of v Figure 13: The three ingredients of JβJ_β. On the left, the three lifted triples equation 33 sitting over one edge uvuv of G, drawn as triangles sharing the palette vertex wβ(uv)w_β(uv). On the upper right, the chain of copies of S that forces w1,…,wtw_1,…,w_t to receive a single common color. On the lower right, the copy of D that forces the three copies of a vertex v to receive three distinct colors. Lemma C.12.4. For every β∈β the hypergraph JβJ_β is a linear triple system, and |V∗|≤C0m|V |≤ C_0m for a constant C0C_0 depending only on S and D. Proof. Write s=|V(S)|s=|V(S)| and d0=|V(D)|d_0=|V(D)|. Counting the vertices of V∗V layer by layer, |V∗|=3m+t+(t−1)(s−2)+m(d0−3)≤(4s+d0−4)m,|V |=3m+t+(t-1)(s-2)+m(d_0-3)≤(4s+d_0-4)\,m, since t=4mt=4m; take C0=4s+d0−4C_0=4s+d_0-4. Every edge of JβJ_β has three vertices, so it remains to check that no pair of vertices lies in two edges. Consider first a pair of the form ui,vi\u^i,v^i\ with u≠vu≠ v. No edge of a gadget copy contains copies of two distinct vertices of G, and a lifted triple containing both uiu^i and viv^i lies over the edge uvuv in layer i, so at most one edge contains the pair. Next consider a pair vi,wj\v^i,w_j\. Gadget copies contribute no such edge, and a lifted triple containing this pair lies over an edge of G incident to v and colored j; since β is proper there is at most one such edge, and the layer i is determined as well. A pair wj,wj′\w_j,w_j \ lies in no lifted triple, and it lies in no gadget edge either, because the copies of S meet each other only in single palette vertices and no edge of S contains both designated vertices. A pair vi,vi′\v^i,v^i \ with i≠i′i≠ i lies in no lifted triple, and the only gadget copy containing it is the copy of D attached to v. A pair ui,vi′\u^i,v^i \ with u≠vu≠ v and i≠i′i≠ i lies in no edge at all: every lifted triple uses a single layer, and no gadget edge contains copies of two distinct vertices of G. Every remaining pair contains a fresh internal vertex, which lies in a single gadget copy, so every edge containing the pair is an edge of that copy; each copy is itself a linear triple system, so there is at most one such edge. Hence no pair of vertices lies in two edges of JβJ_β. ∎ Lemma C.12.5. For every β∈β the hypergraph JβJ_β is not 33-colorable. Proof. Suppose ψ is a proper 33-coloring of JβJ_β. The copies of S give ψ(wj)=ψ(wj+1)ψ(w_j)=ψ(w_j+1) for every j∈[t−1]j∈[t-1] by Lemma C.12.2, so ψ(w1)=ψ(w2)=⋯=ψ(wt),ψ(w_1)=ψ(w_2)=·s=ψ(w_t), and after relabeling the colors we may assume this common value is 11. For each v∈V(G)v∈ V(G) the copy of D attached to v1,v2,v3v^1,v^2,v^3 forces those three vertices to receive three distinct colors, so there is a unique index ϕ(v)∈[3]φ(v)∈[3] with ψ(vϕ(v))=1ψ(v^φ(v))=1. Let uv∈E(G)uv∈ E(G) and suppose ϕ(u)=ϕ(v)=iφ(u)=φ(v)=i. Then the lifted triple ui,vi,wβ(uv)\u^i,v^i,w_β(uv)\ is monochromatic of color 11, a contradiction. Hence ϕφ is a proper 33-coloring of G, which is impossible because G is 44-chromatic. ∎ C.12.3 Recovering the edge coloring For uv∈E(G)uv∈ E(G) write Luv=ui,vi,wβ(uv):i∈[3]L_uv= \\u^i,v^i,w_β(uv)\:i∈[3] \ for the set of three lifted triples over uvuv. These sets are pairwise disjoint as uvuv ranges over E(G)E(G). Lemma C.12.6. Let K⊆JβK J_β be a subhypergraph that is not 33-colorable. Then K contains a triple from LuvL_uv for every uv∈E(G)uv∈ E(G). Proof. Suppose instead that K contains no triple of LuvL_uv for some uv∈E(G)uv∈ E(G), so that K is a subhypergraph of Jβ−LuvJ_β-L_uv. We exhibit a proper 33-coloring of Jβ−LuvJ_β-L_uv, which restricts to one of K. Since G is 44-critical, G−uvG-uv has a proper 33-coloring ϕ:V(G)→[3]φ V(G)→[3]. Give every palette vertex wjw_j the color 11, and for each z∈V(G)z∈ V(G) color z1,z2,z3z^1,z^2,z^3 with the three colors in such a way that zϕ(z)z^φ(z) receives the color 11. Let xi,yi,wβ(xy)\x^i,y^i,w_β(xy)\ be a retained lifted triple, so xy∈E(G)xy∈ E(G) and xy≠uvxy≠ uv. If this triple were monochromatic then, as wβ(xy)w_β(xy) has color 11, we would have ϕ(x)=ϕ(y)=iφ(x)=φ(y)=i, contradicting ϕ(x)≠ϕ(y)φ(x)≠φ(y). So no retained lifted triple is monochromatic. On each copy of S the two designated vertices have received the same color, and on each copy of D the three designated vertices have received three distinct colors, so by Lemma C.12.2 the coloring extends over the fresh internal vertices of every gadget copy. Thus Jβ−LuvJ_β-L_uv is 33-colorable, contradicting the hypothesis on K. ∎ For each β∈β choose an edge-minimal non-33-colorable subhypergraph of JβJ_β and delete its isolated vertices; call the result KβK_β. Such a subhypergraph exists by Lemma C.12.5. Lemma C.12.7. Each KβK_β is a 44-critical linear triple system with at least e(G)e(G) edges. Moreover the map β↦Kβ K_β is injective. Proof. Linearity is inherited from JβJ_β by Lemma C.12.4. By the choice of KβK_β it is not 33-colorable while Kβ−eK_β-e is 33-colorable for every edge e, and it has no isolated vertices. It is 44-colorable: delete an edge e, take a proper 33-coloring of Kβ−eK_β-e, and if e is monochromatic recolor one vertex of e with a fourth color. The recolored vertex then lies in no monochromatic edge, since it is the only vertex of that color, and every other edge is unaffected. Hence KβK_β is 44-chromatic and 44-critical. By Lemma C.12.6 it contains a triple of LuvL_uv for each of the e(G)e(G) edges uvuv of G, and these sets are pairwise disjoint, so KβK_β has at least e(G)e(G) edges. For injectivity, fix uv∈E(G)uv∈ E(G) and read off β(uv)β(uv) from the labeled hypergraph KβK_β as follows. By Lemma C.12.6 some triple of the form ui,vi,wj\u^i,v^i,w_j\ belongs to KβK_β, and as observed in the proof of Lemma C.12.4 the only edges of JβJ_β containing uiu^i and viv^i are the lifted triples over uvuv, all of which use the palette vertex wβ(uv)w_β(uv). So j=β(uv)j=β(uv). Therefore KβK_β determines β, and distinct edge colorings give distinct labeled hypergraphs on V∗V . ∎ C.12.4 Proof of the main theorem Proof of Theorem C.12.1. Let m be large, let G, A and Kβ:β∈\K_β:β \ be as above, and put M=|V∗|M=|V |, so that M≤C0mM≤ C_0m by Lemma C.12.4. By Lemmas C.12.3 and C.12.7 the family Kβ:β∈\K_β:β \ consists of at least (2m)e(G)(2m)^e(G) distinct 44-critical linear triple systems, all labeled inside the single vertex set V∗V . Each has between 11 and M vertices, so there is an integer q≤Mq≤ M such that at least (2m)e(G)M (2m)^e(G)M of them have exactly q vertices. A labeled system inside V∗V isomorphic to a fixed q-vertex system is determined by an injection of the latter’s vertex set into V∗V , so each isomorphism class accounts for at most M(M−1)⋯(M−q+1)≤M(M-1)·s(M-q+1)≤ M^M of them. Therefore T(4,3,2,q)≥(2m)e(G)M⋅M.T(4,3,2,q)\ ≥\ (2m)^e(G)M· M^M. Using e(G)≥am2e(G)≥ am^2 and M≤C0mM≤ C_0m, log((2m)e(G)M⋅M)≥am2log(2m)−(C0m+1)log(C0m)≥a2m2logm ( (2m)^e(G)M· M^M )\ ≥\ a\,m^2 (2m)-(C_0m+1) (C_0m)\ ≥\ a2\,m^2 m for all sufficiently large m. It remains to compare q with m. On one hand q≤M≤C0mq≤ M≤ C_0m. On the other hand a linear triple system on q vertices has at most (q2)/3 q2 /3 edges, since its edges contain three vertex pairs each and no pair is repeated, so Lemma C.12.7 gives q26≥13(q2)≥e(G)≥am2, q^26\ ≥\ 13 q2\ ≥\ e(G)\ ≥\ a\,m^2, whence q≥6amq≥ 6a\,m. Thus q=q(m)=Θ(m)q=q(m)= (m), and in particular q(m)→∞q(m)→∞ as m→∞m→∞, so the integers q(m)q(m) take infinitely many distinct values. Finally m≥q/C0m≥ q/C_0 and logm≥logq−logC0≥12logq m≥ q- C_0≥ 12 q once q≥C02q≥ C_0^2, so T(4,3,2,q)≥exp(a2m2logm)≥exp(a4C02q2logq).T(4,3,2,q)\ ≥\ ( a2m^2 m )\ ≥\ ( a4C_0^2\,q^2 q ). Writing n=q(m)n=q(m) and c=a/(4C02)c=a/(4C_0^2) completes the proof. ∎ Remark C.12.8. The constant c is not optimized here, and the argument settles only the case (k,r,l)=(4,3,2)(k,r,l)=(4,3,2), and only along a sequence of values of n. Any palette of bounded size would give only exp(O(m2)) (O(m^2)) colorings and would not contradict the conjecture. References. Abbott & Liu (1978) Harvey L Abbott and AC Liu. The existence problem for colour critical linear hypergraphs. Acta Mathematica Academiae Scientiarum Hungarica, 32(3):273–282, 1978. Abbott et al. (1980) HL Abbott, A Liu, and Bjarne Toft. The enumeration problem for color critical linear hypergraphs. Journal of Combinatorial Theory, Series B, 29(1):106–115, 1980. Kostochka (2006) Alexandr Kostochka. Color-critical graphs and hypergraphs with few edges: a survey. In More Sets, Graphs and Numbers: A Salute to Vera Sós and András Hajnal, p. 175–197. Springer, 2006. Rödl & Siggers (2006) Vojtech Rödl and M Siggers. Color critical hypergraphs with many edges. Journal of Graph Theory, 53(1):56–74, 2006. Toft (1970) Bjarne Toft. On the maximal number of edges of critical k-chromatic graphs. (No Title), 1970. C.13 Strongly connected digraphs with no small k-kernel This problem asks how small a k-kernel a strongly connected digraph is guaranteed to have. Spiro asked whether |V(G)|/(k+1)+Ok(1)|V(G)|/(k+1)+O_k(1) is always enough once k≥3k≥ 3; we show that it is not, and that the coefficient 1/(k+1)1/(k+1) is itself wrong, the truth lying between 1/k1/k and 1/(k−1)1/(k-1). The digraphs witnessing this are a hub feeding m parallel directed paths that all return through one common tail, long enough to force the hub out of every k-kernel. Two refutations of the same question were published after our run, and the construction below was obtained independently of both. C.13.1 Introduction All digraphs here are finite and have no loops or parallel arcs; directed cycles of length two are allowed. A set X⊆V(G)X V(G) is stable if no arc of G has both ends in X, and for an integer k≥1k≥ 1 a k-kernel of G is a stable set X⊆V(G)X V(G) such that every vertex of G can be joined from X by a directed path of length at most k. Paths of length 00 are allowed, so a k-kernel covers itself. A 11-kernel is a kernel and a 22-kernel is a quasikernel. In a tournament every stable set is a single vertex, so a quasikernel is exactly a king (Post & Zheng 2023). We write κk(G) _k(G) for the minimum size of a k-kernel of G. Not every digraph has a kernel, but every digraph has a quasikernel by the theorem of Chvátal & Lovász 2006, so κk(G) _k(G) is defined for every k≥2k≥ 2. How small a quasikernel one is entitled to expect is the subject of the small quasi-kernel conjecture, that κ2(G)≤|V(G)|/2 _2(G)≤|V(G)|/2 whenever G has no source. It was posed by P. L. Erdős and Székely in 1976 and stated in print by Erdős & Székely 2010; Erdős et al. 2023 survey what is known. After reversing every arc, k-kernels become the (2,k)(2,k)-kernels of Kwaśnik 2006, that is, the stable sets that absorb every vertex within distance k. Spiro 2026 initiated their extremal study and asked what the source-free hypothesis buys when it is strengthened to strong connectivity. If D is a strongly connected digraph and q≥3q≥ 3 is an integer, does there exist a q-kernel Q of D such that |Q|≤|V(D)|q+1+Oq(1)|Q|≤ |V(D)|q+1+O_q(1)? This is Question 7.7 of Spiro 2026, and it is recorded as Conjecture 1.4 by Nguyen et al. 2024, who attribute it to Spiro and leave it open. We write k throughout for the integer called q there. Spiro observes that the coefficient 1/(k+1)1/(k+1) would be best possible, by considering a collection of directed cycles Ck+2C_k+2 all sharing a single vertex. The restriction to k≥3k≥ 3 is necessary, and is what distinguishes the published form of the question from the first preprint version, which asked it for k≥2k≥ 2: Example 17 of Erdős et al. 2023 exhibits strongly connected digraphs whose smallest quasikernel has size (12−o(1))|V(G)|( 12-o(1))|V(G)|, which is far above |V(G)|/3|V(G)|/3. On the positive side, Spiro 2026 showed that under the hypotheses of the question there is always a k-kernel of size at most about |V(G)|/logk|V(G)|/ k, by producing about logk k pairwise disjoint k-kernels. This was improved by Nguyen et al. 2024, who proved that every digraph G with |V(G)|>1|V(G)|>1 admitting a spanning out-arborescence, and in particular every strongly connected digraph, satisfies κk(G)≤ 1+|V(G)|−2k−1(k≥2). _k(G)\ ≤\ 1+ |V(G)|-2k-1 (k≥ 2). (34) They deduce equation 34 from the acyclic case, where the coefficient improves to 1/k1/k: an acyclic digraph on at least two vertices with only one source has a k-kernel of size at most 1+(|V(G)|−2)/k1+(|V(G)|-2)/k. The digraph of their Figure 1, a source y with a single arc to a hub x together with m disjoint directed paths on k vertices leaving x, shows that this is tight. We show that the answer to Question 7.7 is negative for every k≥3k≥ 3, and that it fails by a linear rather than a constant margin. Theorem C.13.1. For every integer k≥2k≥ 2 and every integer m≥1m≥ 1 there is a strongly connected digraph Gk,mG_k,m on mk+k+2mk+k+2 vertices whose smallest k-kernel has size exactly κk(Gk,m)=m+1=|V(Gk,m)|−2k. _k(G_k,m)\ =\ m+1\ =\ |V(G_k,m)|-2k. Consequently, for every k≥2k≥ 2 and every constant c there is a strongly connected digraph G with κk(G)>|V(G)|/(k+1)+c _k(G)>|V(G)|/(k+1)+c. Theorem C.13.1 does more than defeat the additive term Ok(1)O_k(1): it shows that the coefficient 1/(k+1)1/(k+1) is itself wrong. Write ck∗c_k^* for the infimum of the constants c for which κk(G)≤c|V(G)|+Ok(1) _k(G)≤ c|V(G)|+O_k(1) holds for all strongly connected G. Theorem C.13.1 gives ck∗≥1/kc_k^*≥ 1/k and equation 34 gives ck∗≤1/(k−1)c_k^*≤ 1/(k-1). For k≥3k≥ 3, whether the value 1/k1/k is itself admissible is asked by Wang et al. 2026; for k=2k=2 the corresponding statement c2∗=1/2c_2^*=1/2 would follow from the small quasi-kernel conjecture, since every strongly connected digraph is source-free. The excess is already substantial for moderate parameters: κ3(G3,50)=51 _3(G_3,50)=51 on 155155 vertices against 155/4=38.75155/4=38.75, and κ10(G10,50)=51 _10(G_10,50)=51 on 512512 vertices against 512/11≈46.55512/11≈ 46.55. Taking k=2k=2 gives strongly connected digraphs whose smallest quasikernel has size exactly 12|V(G)|−1 12|V(G)|-1, one less than the bound of the small quasi-kernel conjecture; digraphs achieving this up to o(|V(G)|)o(|V(G)|) were already given by Erdős et al. 2023. Two refutations have since appeared. Wang et al. 2026 give for each k≥3k≥ 3 strongly connected digraphs every k-kernel of which has size at least (|V(G)|−2)/k(|V(G)|-2)/k; this is the bound of Theorem C.13.1, which also covers k=2k=2. Penev et al. 2026 obtain the weaker bound κk(G)>(1/k−ε)|V(G)| _k(G)>(1/k- )|V(G)|, but with the extra feature that the digraphs may be taken of arbitrarily large directed girth. By contrast, every directed cycle of Gk,mG_k,m passes through the hub x and hence traverses a whole arm, so these cycles have exactly the lengths k+2,k+3,…,2k+2k+2,k+3,…,2k+2 and Gk,mG_k,m has directed girth only k+2k+2. The same paper shows that for odd k≥3k≥ 3 every source-free bipartite digraph with a spanning out-arborescence, and in particular every strongly connected bipartite digraph, has a k-kernel of size at most |V(G)|/(k+1)+1|V(G)|/(k+1)+1. For odd k, then, no bipartite digraph can refute Question 7.7, and indeed the lengths just listed include odd ones, so Gk,mG_k,m is not bipartite. The construction is the acyclic example of Figure 1 of Nguyen et al. 2024 with a return path attached: the m parallel directed paths leaving the hub x, each on k vertices, are brought back to x through one common tail. In the acyclic example the hub is unusable because the source y lies in every k-kernel and sends an arc to x; here y is no longer a source, and the tail takes over that role. The tail is long enough that some vertex of it must be used to cover its own far end, and every vertex of the tail sends an arc to the hub; this forces the hub out of every k-kernel and thereby forces each of the m paths to pay for itself. C.13.2 The construction Fix integers k≥2k≥ 2 and m≥1m≥ 1. The digraph Gk,mG_k,m has vertex set V(Gk,m)=x,y,p1,…,pk∪ai,j:1≤i≤m, 1≤j≤kV(G_k,m)=\x,y,p_1,…,p_k\\ ∪\ \a_i,j:1≤ i≤ m,\ 1≤ j≤ k\ and arc set consisting of y→x,pj→x(1≤j≤k),pj→pj+1(1≤j<k),pk→y,y→ x, p_j→ x\ \ (1≤ j≤ k), p_j→ p_j+1\ \ (1≤ j<k), p_k→ y, together with, for each i∈1,…,mi∈\1,…,m\, the arm x→ai,1→ai,2→⋯→ai,k→p1.x→ a_i,1→ a_i,2→·s→ a_i,k→ p_1. There are no other arcs. Write Ai:=ai,1,…,ai,kA_i:=\a_i,1,…,a_i,k\ for the iith arm and P:=y,p1,…,pkP:=\y,p_1,…,p_k\ for the tail. Thus V(Gk,m)V(G_k,m) is the disjoint union of x\x\, P and A1,…,AmA_1,…,A_m, and |V(Gk,m)|=1+(k+1)+mk=mk+k+2.|V(G_k,m)|=1+(k+1)+mk=mk+k+2. (35) Figure 14 shows the digraph. xxa1,1a_1,1a1,2a_1,2⋯·sa1,ka_1,k⋮ ,1a_m,1am,2a_m,2⋯·sam,ka_m,kp1p_1p2p_2⋯·spkp_ky Figure 14: The digraph Gk,mG_k,m. The hub x feeds m arms, each a directed path with k internal vertices running from x to p1p_1, and the tail p1→p2→⋯→pk→y→xp_1→ p_2→·s→ p_k→ y→ x closes the digraph up. The dashed arcs pj→xp_j→ x, present for every j, are what keep x out of every k-kernel. Lemma C.13.2. Gk,mG_k,m is strongly connected. Proof. Every vertex reaches x: the vertex y and each pjp_j do so along a single arc, and ai,ja_i,j reaches p1p_1 along its own arm and then uses p1→xp_1→ x. Conversely x reaches every vertex: it reaches all of AiA_i along the iith arm and then p1p_1, then p2,…,pkp_2,…,p_k along the tail, and finally y. Hence any vertex reaches any other through x. ∎ C.13.3 A small k-kernel We first exhibit a k-kernel of the size claimed in Theorem C.13.1. Lemma C.13.3. The set K0:=y∪ai,k:1≤i≤mK_0:=\y\∪\a_i,k:1≤ i≤ m\ is a k-kernel of Gk,mG_k,m, and |K0|=m+1|K_0|=m+1. Proof. First, K0K_0 is stable. Indeed the only arcs incident with y are y→xy→ x and pk→yp_k→ y, and the only arcs incident with ai,ka_i,k are ai,k−1→ai,ka_i,k-1→ a_i,k and ai,k→p1a_i,k→ p_1; none of x, pkp_k, p1p_1, ai,k−1a_i,k-1 lies in K0K_0. In particular distinct vertices ai,ka_i,k and ai′,ka_i ,k are nonadjacent, as they lie on different arms. For the covering condition, y→xy→ x gives dist(y,x)=1dist(y,x)=1, and y→x→ai,1→⋯→ai,jy→ x→ a_i,1→·s→ a_i,j is a directed path of length j+1j+1, so dist(y,ai,j)≤kdist(y,a_i,j)≤ k for every j≤k−1j≤ k-1. Each ai,ka_i,k covers itself, and ai,k→p1→p2→⋯→pja_i,k→ p_1→ p_2→·s→ p_j has length j≤kj≤ k, so any one ai,ka_i,k covers all of p1,…,pkp_1,…,p_k. Every vertex of Gk,mG_k,m is therefore joined from K0K_0 by a directed path of length at most k. ∎ C.13.4 Every k-kernel is large We now show that no k-kernel of Gk,mG_k,m can be smaller. Lemma C.13.4. Every k-kernel K of Gk,mG_k,m satisfies |K|≥m+1|K|≥ m+1. Proof. The crucial observation is that the far end of the tail and the far end of each arm can each be covered from only one place. First consider y. The unique in-neighbor of y is pkp_k, the unique in-neighbor of pjp_j is pj−1p_j-1 for 2≤j≤k2≤ j≤ k, and the in-neighbors of p1p_1 are the m vertices ai,ka_i,k. Walking backwards from y therefore gives dist(pk+1−ℓ,y)=ℓdist(p_k+1- ,y)= for 1≤ℓ≤k1≤ ≤ k, while every remaining vertex is at distance at least k+1k+1 from y. So the vertices u with dist(u,y)≤kdist(u,y)≤ k are exactly the k+1k+1 vertices of P, and hence K∩P≠∅K∩ P≠ . Next, every vertex of P has an arc to x, by y→xy→ x and pj→xp_j→ x. Picking u∈K∩Pu∈ K∩ P, the pair u,xu,x is joined by an arc, so stability of K forces x∉Kx∉ K. Now fix i and consider ai,ka_i,k. The unique in-neighbor of ai,ja_i,j is ai,j−1a_i,j-1 for 2≤j≤k2≤ j≤ k, and the unique in-neighbor of ai,1a_i,1 is x. Walking backwards from ai,ka_i,k therefore gives dist(ai,k−ℓ,ai,k)=ℓdist(a_i,k- ,a_i,k)= for 1≤ℓ≤k−11≤ ≤ k-1 and dist(x,ai,k)=kdist(x,a_i,k)=k, while every remaining vertex is at distance at least k+1k+1 from ai,ka_i,k. So the vertices u with dist(u,ai,k)≤kdist(u,a_i,k)≤ k are exactly the k+1k+1 vertices of Ai∪xA_i∪\x\, and since x∉Kx∉ K we get K∩Ai≠∅K∩ A_i≠ . The sets P,A1,…,AmP,A_1,…,A_m are pairwise disjoint, so |K|≥m+1|K|≥ m+1. ∎ C.13.5 Proof of the main theorem Proof of Theorem C.13.1. The digraph Gk,mG_k,m is strongly connected by Lemma C.13.2 and has mk+k+2mk+k+2 vertices by equation 35. Lemmas C.13.3 and C.13.4 give κk(Gk,m)=m+1 _k(G_k,m)=m+1, and m+1=mk+k=|V(Gk,m)|−2km+1= mk+kk= |V(G_k,m)|-2k by equation 35. For the last assertion, fix k≥2k≥ 2 and compute κk(Gk,m)−|V(Gk,m)|k+1=(m+1)−mk+k+2k+1=(m+1)(k+1)−mk−k−2k+1=m−1k+1. _k(G_k,m)- |V(G_k,m)|k+1=(m+1)- mk+k+2k+1= (m+1)(k+1)-mk-k-2k+1= m-1k+1. This tends to infinity as m→∞m→∞, so given c we may choose m with (m−1)/(k+1)>c(m-1)/(k+1)>c and take G=Gk,mG=G_k,m. ∎ Remark C.13.5. The hub-and-arms mechanism is not tied to the particular tail used above. Fix integers k≥2k≥ 2 and t≥1t≥ 1, and write C=c0,…,ckC=\c_0,…,c_k\, R=r1,…,rk−1R=\r_1,…,r_k-1\ and Bi=bi,1,…,bi,kB_i=\b_i,1,…,b_i,k\ for 1≤i≤t1≤ i≤ t. The digraph Hk,tH_k,t has vertex set V(Hk,t)=C∪x,z∪R∪B1∪⋯∪BtV(H_k,t)=C∪\x,z\∪ R∪ B_1∪·s∪ B_t and arc set consisting of c0→c1→⋯→ck→c0,cj→x(0≤j≤k),z→r1→⋯→rk−1→c0,c_0→ c_1→·s→ c_k→ c_0, c_j→ x\ \ (0≤ j≤ k), z→ r_1→·s→ r_k-1→ c_0, together with, for each i∈1,…,ti∈\1,…,t\, the arm x→bi,1→⋯→bi,k→z.x→ b_i,1→·s→ b_i,k→ z. Then Hk,tH_k,t is strongly connected on kt+2k+2kt+2k+2 vertices, and κk(Hk,t)=t+1 _k(H_k,t)=t+1. The upper bound comes from the k-kernel c0∪bi,k:1≤i≤t\c_0\∪\b_i,k:1≤ i≤ t\, and the lower bound repeats the proof of Lemma C.13.4. Indeed the only arc entering C from outside is rk−1→c0r_k-1→ c_0, so a path from outside C to ckc_k has length at least 1+dist(c0,ck)=k+11+dist(c_0,c_k)=k+1 and ckc_k is covered only from within C; every cjc_j has an arc to x, so x lies in no k-kernel; and the only arc entering BiB_i from outside is x→bi,1x→ b_i,1, so bi,kb_i,k is covered only from Bi∪xB_i∪\x\ and every k-kernel meets every BiB_i. Since t+1=(|V(Hk,t)|−k−2)/kt+1=(|V(H_k,t)|-k-2)/k, this family gives the same coefficient 1/k1/k as Theorem C.13.1 but loses 11 in the additive constant. References. Chvátal & Lovász (2006) Vašek Chvátal and László Lovász. Every directed graph has a semi-kernel. In Hypergraph Seminar: Ohio State University 1972, p. 175–175. Springer, 2006. Erdős & Székely (2010) Péter L Erdős and László A Székely. Two conjectures on quasi-kernels. In Fete of Combinatorics and Computer Science, volume 20 of Bolyai Society Mathematical Studies, p. 357–358. Springer, 2010. Open problems no. 4. Erdős et al. (2023) Péter L Erdős, Ervin Győri, Tamás Róbert Mezei, Nika Salia, and Mykhaylo Tyomkyn. On the small quasi-kernel conjecture. arXiv preprint arXiv:2307.04112, 2023. Kwaśnik (2006) Maria Kwaśnik. On the (k; l)-kernels. In Graph Theory: Proceedings of a Conference held in Łagów, Poland, February 10–13, 1981, p. 114–121. Springer, 2006. Nguyen et al. (2024) Tung Nguyen, Alex Scott, and Paul Seymour. Distant digraph domination. arXiv preprint arXiv:2409.05039, 2024. Penev et al. (2026) Irena Penev, Maya Stein, and Ana Trujillo-Negrete. Small q-kernels in digraphs. arXiv preprint arXiv:2608.00825, 2026. Post & Zheng (2023) Logan Post and Zeyu Zheng. Common kings of a chain of cycles in a strong tournament. Graphs and Combinatorics, 39(4):71, 2023. Spiro (2026) Sam Spiro. Generalized quasikernels in digraphs. European Journal of Combinatorics, 133:104307, 2026. Wang et al. (2026) Xiaoyi Wang, Bo Deng, and Bin Chen. Counterexamples to a problem of spiro on k-kernels. Discrete Mathematics, 349(12):115303, 2026. C.14 F-positivity of chromatic symmetric functions of hypertrees The chromatic symmetric function of a hypergraph is the generating function for the colorings under which no hyperedge is monochromatic. Unlike its graph analogue, it need not expand nonnegatively in Gessel’s fundamental quasisymmetric basis, but Taylor 2015 proved that it does whenever the hypergraph is a hypertree all of whose hyperedges have prime size, and conjectured that primality is superfluous. We prove the conjecture. The prime case rests on partitioning the nonconstant colorings of an r-element hyperedge into r!r! blocks indexed by the linear orders of that hyperedge, and no general construction of such a partition is known. We replace the partition by a decomposition of the indicator function of “nonconstant” into chain conditions with nonnegative rational weights; such a decomposition exists for every r, and it glues across a hypertree exactly as a partition would. C.14.1 Introduction A hypergraph is a pair H=(V,E)H=(V,E) with V finite and E a family of subsets of V, each of size at least 22, called hyperedges. A coloring of H is a map κ:V→ℕκ:V , where ℕ=1,2,…N=\1,2,…\, and κ is proper if no hyperedge is monochromatic, that is, if κ is nonconstant on every e∈Ee∈ E. The chromatic symmetric function of H is XH=∑κproper∏v∈Vxκ(v),X_H= _κ\ proper\ _v∈ Vx_κ(v), introduced for ordinary graphs by Stanley 1995 and extended to hypergraphs by Stanley 1998. Write n=|V|n=|V| and [n]=1,…,n[n]=\1,…,n\. For S⊆[n−1]S [n-1] the fundamental quasisymmetric function of Gessel 1984 is FS(n)=∑i1≤⋯≤inj∈S⇒ij<ij+1xi1⋯xin,F_S^(n)= _ subarrayci_1≤·s≤ i_n\\ j∈ S i_j<i_j+1 subarrayx_i_1·s x_i_n, and the FS(n)F_S^(n) with S⊆[n−1]S [n-1] form a basis of the degree-n component of the ring QSymQSym of quasisymmetric functions. A homogeneous quasisymmetric function of degree n is F-positive if all of its coefficients in this basis are nonnegative. A path in H is a sequence v1,e1,v2,e2,…,em,vm+1v_1,e_1,v_2,e_2,…,e_m,v_m+1 with vi,vi+1∈eiv_i,v_i+1∈ e_i for each i, in which the hyperedges eie_i are distinct and the vertices viv_i are distinct except that v1=vm+1v_1=v_m+1 is allowed. It is a cycle if v1=vm+1v_1=v_m+1 and m≥2m≥ 2. The hypergraph H is connected if any two vertices are joined by a path, and a hypertree is a connected hypergraph with no cycles; this is the convention of Gessel & Kalikow 2005 adopted by Taylor 2015. For an ordinary graph, XGX_G is always F-positive: Stanley 1995 splits the proper colorings according to the acyclic orientation each one induces and identifies every piece as a (P,ω)(P,ω)-partition enumerator. For hypergraphs this fails. Taylor 2015 records that the hypergraph on V=[4]V=[4] with hyperedges 1,2,3\1,2,3\ and 2,3,4\2,3,4\ has XH=2F1+6F2+2F3+4F1,2+8F1,3+4F2,3−2F1,2,3,X_H=2F_\1\+6F_\2\+2F_\3\+4F_\1,2\+8F_\1,3\+4F_\2,3\-2F_\1,2,3\, and that the hypergraph with hyperedges 1,2,3,1,4,2,4,3,4,5\1,2,3\,\1,4\,\2,4\,\3,4,5\ is not F-positive either, although distinct hyperedges of it meet in at most one vertex. Both of these contain cycles. On the other hand Taylor 2015 proves that XHX_H is F-positive whenever H is a hypertree each of whose hyperedges has prime size, and in that case obtains the explicit expansion XH=∑πFDesH(π)(n)X_H= _πF^(n)_Des_H(π) over all n!n! bijections π:V→[n]π:V→[n], where the H-descents DesH(π)Des_H(π) are read off from the unique path between consecutively labeled vertices. Taylor 2015 asks for the removal of the primality hypothesis: Let H be a hypertree. Then XHX_H is F-positive. The role of primality is isolated by Taylor 2015. Call an integer r≥2r≥ 2 splittable if the nonconstant colorings of an r-element set can be partitioned into r!r! blocks indexed by the linear orders of that set, the block of a linear order consisting of the colorings that are weakly increasing along it and strictly increasing at some prescribed set of steps; he proves that XHX_H is F-positive for every hypertree all of whose hyperedge sizes are splittable. He shows that every prime is splittable, by the cyclic standardization of Gessel & Reutenauer 1993, and reports a splitting for r=4r=4 found by computer search; he also observes that splittability of r is equivalent to partitionability of the Coxeter complex of type Ar−1A_r-1 with the empty face removed, a scheduling problem in the sense of Breuer & Klivans 2016. Conjecture A was therefore known for every hypertree whose hyperedge sizes are all prime or equal to 44, and in particular for every hypertree with no hyperedge on more than 55 vertices. Each further composite size requires a splitting of its own. In a different direction, Pawlowski 2018 proves that XHX_H is Schur-positive, hence F-positive, for every hyperforest whose line graph is bipartite, which settles Conjecture B of Taylor 2015; this does not reach all hypertrees, since three 66-element hyperedges through a common vertex form a hypertree whose line graph is K3K_3 and whose hyperedge sizes are neither prime nor 44. Theorem C.14.1. Let H=(V,E)H=(V,E) be a finite hypertree and n=|V|n=|V|. Then XH=∑S⊆[n−1]aSFS(n)with aS∈ℤ≥0for every S⊆[n−1].X_H= _S [n-1]a_S\,F_S^(n) a_S _≥ 0\ for every S [n-1]. This is Conjecture A of Taylor 2015. Every bijection V→[n]V→[n] is a proper coloring, and the coefficient of x1x2⋯xnx_1x_2·s x_n in FS(n)F_S^(n) equals 11 for every S, so the coefficients of Theorem C.14.1 satisfy ∑S⊆[n−1]aS=n! _S [n-1]a_S=n!, as Taylor 2015 observes. Theorem C.14.1 therefore says that XHX_H is a sum of n!n! fundamental quasisymmetric functions counted with multiplicity. Certain cases are immediate. If n=1n=1 then E=∅E= , since every hyperedge has at least two vertices, and XH=x1+x2+⋯=F∅(1)X_H=x_1+x_2+·s=F_ ^(1). We assume n≥2n≥ 2 from now on, so that every vertex of H lies in a hyperedge. We now summarize the proof. Fix a hyperedge e of size r and, for a linear order τ of e and a set B⊆[r−1]B [r-1], consider the colorings of e that are weakly increasing along τ and strictly increasing at the steps in B. The key point is that the indicator function of “κ is nonconstant on e” is a nonnegative rational combination of the indicator functions of these conditions, taken over all r!r! orders τ and all nonempty B, with weights depending only on r and B. The weights are forced by the fundamental expansion of the chromatic symmetric function of a single hyperedge, and their nonnegativity is the elementary fact that every subset of [r−1][r-1] is the descent set of at least one permutation of [r][r]. Multiplying this local identity over the hyperedges expresses XHX_H as a nonnegative rational combination of generating functions for systems of weak and strict inequalities, one inequality per consecutive pair inside each chosen order. The hypertree hypothesis makes each such system a (P,ω)(P,ω)-partition condition on an ordinary tree, so each of these generating functions is F-positive, and the fundamental coefficients of XHX_H are nonnegative rationals. They are integers because the monomial quasisymmetric coefficients of XHX_H are integers and the two bases are related by an integral unitriangular matrix. C.14.2 A weighted local decomposition for one hyperedge Fix r≥2r≥ 2. For π∈rπ∈ S_r write Des(π)=j∈[r−1]:π(j)>π(j+1)Des(π)=\j∈[r-1]:π(j)>π(j+1)\, and for B⊆[r−1]B [r-1] put Ar(B)=#π∈r:Des(π)=B,wr(B)=Ar(B)−(−1)|B|r!.A_r(B)=\#\π∈ S_r:Des(π)=B\, w_r(B)= A_r(B)-(-1)^|B|r!. Since Ar(∅)=1A_r( )=1 we have wr(∅)=0w_r( )=0, so the empty set never contributes below. Lemma C.14.2. For every B⊆[r−1]B [r-1] one has wr(B)≥0w_r(B)≥ 0. Proof. Every subset of [r−1][r-1] is the descent set of at least one permutation in r S_r. Indeed, let B cut [r][r] into consecutive blocks, fill the first block increasingly with the largest available values of [r][r], the second block increasingly with the next largest available values, and so on. This permutation ascends inside each block and descends exactly at the cuts, so its descent set is B and Ar(B)≥1A_r(B)≥ 1. If |B||B| is odd then r!wr(B)=Ar(B)+1>0r!\,w_r(B)=A_r(B)+1>0, and if |B||B| is even then r!wr(B)=Ar(B)−1≥0r!\,w_r(B)=A_r(B)-1≥ 0. ∎ For P=p1<⋯<pk⊆[r−1]P=\p_1<·s<p_k\ [r-1] let α(P)=(α1(P),…,αk+1(P))=(p1,p2−p1,…,pk−pk−1,r−pk)α(P)=( _1(P),…, _k+1(P))=(p_1,\,p_2-p_1,\,…,\,p_k-p_k-1,\,r-p_k) be the composition of r whose partial sums are the elements of P, with α(∅)=(r)α( )=(r). Lemma C.14.3. If ∅≠P⊆[r−1] ≠ P [r-1] then ∑∅≠B⊆Pwr(B)=1∏iαi(P)!. _ ≠ B Pw_r(B)= 1 _i _i(P)!. Proof. A permutation of [r][r] has descent set contained in P exactly when it is increasing on each of the blocks that P cuts [r][r] into, and such a permutation is determined by the unordered choice of values placed in those blocks. Hence ∑B⊆PAr(B)=r!∏iαi(P)!. _B PA_r(B)= r! _i _i(P)!. Since P≠∅P≠ we also have ∑B⊆P(−1)|B|=0 _B P(-1)^|B|=0, and therefore ∑B⊆Pwr(B)=1r!(r!∏iαi(P)!−0)=1∏iαi(P)!. _B Pw_r(B)= 1r! ( r! _i _i(P)!-0 )= 1 _i _i(P)!. The summand wr(∅)w_r( ) vanishes, so it may be omitted. ∎ Let e be a set of size r. For a linear order τ=(u1,…,ur)τ=(u_1,…,u_r) of the elements of e and a set B⊆[r−1]B [r-1], let C(τ,B)C(τ,B) be the set of colorings κ:e→ℕκ:e with κ(u1)≤⋯≤κ(ur),κ(uj)<κ(uj+1) for every j∈B.κ(u_1)≤·s≤κ(u_r), κ(u_j)<κ(u_j+1)\ for every j∈ B. Proposition C.14.4. For every coloring κ:e→ℕκ:e , κis nonconstant one=∑τ∑∅≠B⊆[r−1]wr(B) 1κ∈C(τ,B),1_\κ\ is nonconstant on\ e\= _τ\ _ ≠ B [r-1]w_r(B)\,1_\κ∈ C(τ,B)\, where τ runs over all r!r! linear orders of e. Proof. Suppose first that κ is constant. Then no strict inequality holds, so κ∉C(τ,B)κ∉ C(τ,B) for every τ and every nonempty B, and the right hand side is 00. Suppose now that κ is nonconstant, and let the fibers of κ have sizes α1,…,αk _1,…, _k listed in increasing order of color, so that k≥2k≥ 2 and α1+⋯+αk=r _1+·s+ _k=r. There are exactly ∏iαi! _i _i! linear orders τ=(u1,…,ur)τ=(u_1,…,u_r) along which κ is weakly increasing, namely those obtained by listing the fibers in increasing order of color and ordering each fiber arbitrarily. For any other order τ the coloring κ lies in no C(τ,B)C(τ,B) at all. Fix one of the ∏iαi! _i _i! weakly increasing orders. Along it the strict jumps occur exactly at the set P=α1,α1+α2,…,α1+⋯+αk−1,P=\ _1,\ _1+ _2,\ …,\ _1+·s+ _k-1\, which is nonempty because k≥2k≥ 2, and α(P)=(α1,…,αk)α(P)=( _1,…, _k). Hence κ∈C(τ,B)κ∈ C(τ,B) if and only if B⊆PB P, so the inner sum for this τ equals ∑∅≠B⊆Pwr(B)=1∏iαi! _ ≠ B Pw_r(B)= 1 _i _i! by Lemma C.14.3. Summing over the ∏iαi! _i _i! weakly increasing orders gives 11, as desired. ∎ Example C.14.5. Take r=4r=4, the smallest hyperedge size not covered by primality. The descent-set counts on 4 S_4 are A4(∅)=A4(1,2,3)=1A_4( )=A_4(\1,2,3\)=1, A4(1)=A4(3)=A4(1,2)=A4(2,3)=3A_4(\1\)=A_4(\3\)=A_4(\1,2\)=A_4(\2,3\)=3 and A4(2)=A4(1,3)=5A_4(\2\)=A_4(\1,3\)=5, so w4(1)=w4(3)=w4(1,3)=16,w4(2)=14,w_4(\1\)=w_4(\3\)=w_4(\1,3\)= 16, w_4(\2\)= 14, w4(1,2)=w4(2,3)=w4(1,2,3)=112.w_4(\1,2\)=w_4(\2,3\)=w_4(\1,2,3\)= 112. If κ takes two distinct values on e, each twice, then ∏iαi!=4 _i _i!=4 orders are weakly increasing, each with P=2P=\2\, and the total contribution is 4⋅w4(2)=14· w_4(\2\)=1. If κ is injective then a single order is weakly increasing, with P=1,2,3P=\1,2,3\, and the total contribution is 16+14+16+112+16+112+112=1 16+ 14+ 16+ 112+ 16+ 112+ 112=1. C.14.3 Gluing the local identities over a hypertree For each hyperedge e choose a linear order τe=(ue,1,…,ue,|e|) _e=(u_e,1,…,u_e,|e|) of its vertices together with a nonempty set Be⊆[|e|−1]B_e [|e|-1], and let Ω denote such a collection of choices. Put WΩ=∏e∈Ew|e|(Be),W_ = _e∈ Ew_|e|(B_e), which is nonnegative by Lemma C.14.2, and let KΩK_ be the generating function ∑κ∏v∈Vxκ(v) _κ _v∈ Vx_κ(v) over the colorings κ:V→ℕκ:V satisfying, for every e∈Ee∈ E, κ(ue,1)≤⋯≤κ(ue,|e|),κ(ue,j)<κ(ue,j+1) for every j∈Be.κ(u_e,1)≤·s≤κ(u_e,|e|), κ(u_e,j)<κ(u_e,j+1)\ for every j∈ B_e. (36) A coloring is proper exactly when it is nonconstant on every hyperedge, so multiplying the identity of Proposition C.14.4 over the hyperedges of H gives the pointwise identity κproper=∑ΩWΩ 1κsatisfies the inequalities ofΩ,1_\κ\ proper\= _ W_ \,1_\κ\ satisfies the inequalities of\ \, in which “the inequalities of Ω ” abbreviates equation 36 for every e∈Ee∈ E and the sum over Ω is finite. Weighting by ∏v∈Vxκ(v) _v∈ Vx_κ(v) and summing over all colorings therefore gives XH=∑ΩWΩKΩ.X_H= _ W_ K_ . (37) We now use the hypertree hypothesis to identify each KΩK_ as a (P,ω)(P,ω)-partition enumerator. Lemma C.14.6. Let H=(V,E)H=(V,E) be a hypertree with |V|≥2|V|≥ 2. Then its incidence graph is a tree, distinct hyperedges of H meet in at most one vertex, and ∑e∈E(|e|−1)=|V|−1. _e∈ E(|e|-1)=|V|-1. Proof. The incidence graph has vertex set V⊔EV E and an edge joining v to e whenever v∈ev∈ e. Since |V|≥2|V|≥ 2 and H is connected, every vertex of H lies in a hyperedge, and every path of H from v to v′v is a walk of the incidence graph from v to v′v ; as each hyperedge is adjacent to its own vertices, the incidence graph is connected. It is bipartite and simple, so each of its cycles has even length 2m2m with m≥2m≥ 2 and reads v1,e1,v2,e2,…,vm,em,v1v_1,e_1,v_2,e_2,…,v_m,e_m,v_1 with the viv_i distinct and the eie_i distinct, which is precisely a cycle of H. As H has no cycles the incidence graph is acyclic, hence a tree. A tree on |V|+|E||V|+|E| vertices has |V|+|E|−1|V|+|E|-1 edges, and the incidence graph has ∑e∈E|e| _e∈ E|e| edges, so ∑e∈E|e|=|V|+|E|−1 _e∈ E|e|=|V|+|E|-1, which is the displayed identity. Finally, if distinct hyperedges e,e′e,e both contained distinct vertices u,vu,v, then u,e,v,e′,u,e,v,e ,u would be a cycle of H. ∎ Given Ω , let TΩT_ be the graph on vertex set V whose edges join consecutive vertices in the chosen order of each hyperedge, E(TΩ)=ue,j,ue,j+1:e∈E, 1≤j<|e|.E(T_ )= \\u_e,j,u_e,j+1\\ :\ e∈ E,\ 1≤ j<|e| \. Lemma C.14.7. For every Ω the graph TΩT_ is a tree on V. Proof. It is connected. Indeed, let u,v∈Vu,v∈ V be distinct and let u=v1,e1,…,em,vm+1=vu=v_1,e_1,…,e_m,v_m+1=v be a path of H. For each i the vertices viv_i and vi+1v_i+1 both lie in eie_i, and the edges of TΩT_ contributed by eie_i form a path through all of eie_i, so viv_i and vi+1v_i+1 are joined in TΩT_ . Next, the ∑e∈E(|e|−1) _e∈ E(|e|-1) listed pairs are pairwise distinct: consecutive pairs inside one linear order are distinct, and two pairs coming from different hyperedges are distinct because distinct hyperedges share at most one vertex by Lemma C.14.6. Hence TΩT_ is a connected graph on |V||V| vertices with ∑e∈E(|e|−1)=|V|−1 _e∈ E(|e|-1)=|V|-1 edges, again by Lemma C.14.6, and is therefore a tree. ∎ Orient every edge of TΩT_ as ue,j→ue,j+1u_e,j→ u_e,j+1, and call this oriented edge strict if j∈Bej∈ B_e and weak otherwise. By Lemma C.14.7 the underlying graph is a tree, so this orientation is acyclic and the transitive closure of these relations is a partial order; let PΩP_ be the resulting poset on V, generated by the relations ue,j<PΩue,j+1u_e,j<_P_ u_e,j+1. Let QΩQ_ be the second orientation of the same tree obtained by keeping every weak edge and reversing every strict edge. It is again acyclic, so we may choose a bijection ωΩ:V→[n] _ :V→[n] that is increasing along QΩQ_ ; equivalently, along an oriented edge u→vu→ v of TΩT_ , ωΩ(u)<ωΩ(v) if the edge is weak,ωΩ(u)>ωΩ(v) if the edge is strict. _ (u)< _ (v)\ if the edge is weak, _ (u)> _ (v)\ if the edge is strict. (38) Figure 15 shows the construction on a small hypertree. 112233445566e1e_1e2e_2the hypertree H112233445566the oriented tree TΩT_ Figure 15: On the left, the hypertree H on V=[6]V=[6] with hyperedges e1=1,2,3e_1=\1,2,3\ and e2=3,4,5,6e_2=\3,4,5,6\ drawn as gray ovals. On the right, the tree TΩT_ for the choice τe1=(1,2,3) _e_1=(1,2,3), τe2=(3,4,5,6) _e_2=(3,4,5,6), Be1=2B_e_1=\2\ and Be2=1,3B_e_2=\1,3\: each hyperedge is threaded along its chosen order, and the three bold arrows are the strict steps, the two thin gray arrows the weak ones. Reversing the bold arrows gives the acyclic orientation QΩQ_ , and the labeling ωΩ _ that assigns 1,2,3,4,5,61,2,3,4,5,6 to the vertices 4,6,1,3,5,24,6,1,3,5,2 in this order is increasing along it, as required by equation 38. We use the order-preserving convention for P-partitions. Given a finite poset P on n elements and a bijection ω:P→[n]ω:P→[n], a (P,ω)(P,ω)-partition is a map σ:P→ℕσ:P such that x<Pyx<_Py implies σ(x)≤σ(y)σ(x)≤σ(y), and implies σ(x)<σ(y)σ(x)<σ(y) when moreover ω(x)>ω(y)ω(x)>ω(y). Write Γ(P,ω)=∑σ∏v∈Pxσ(v) (P,ω)= _σ _v∈ Px_σ(v) for the generating function of the (P,ω)(P,ω)-partitions. The fundamental theorem of (P,ω)(P,ω)-partitions, for which see Gessel 1984 or, in the order-reversing convention, Stanley 2011, states that Γ(P,ω)=∑π∈ℒ(P)FDω(π)(n),Dω(π)=i∈[n−1]:ω(πi)>ω(πi+1), (P,ω)= _π (P)F^(n)_D_ω(π), D_ω(π)=\i∈[n-1]:ω( _i)>ω( _i+1)\, (39) where ℒ(P)L(P) is the set of linear extensions of P, each written as the word π1π2⋯πn _1 _2·s _n that lists the elements of P in the corresponding order. In particular Γ(P,ω) (P,ω) is F-positive with integer coefficients. Remark C.14.8. The weights of Proposition C.14.4 are forced by the shape of the decomposition. Let pi=x1i+x2i+⋯p_i=x_1^i+x_2^i+·s be the iith power sum symmetric function. Multiplying the identity of Proposition C.14.4 by ∏v∈exκ(v) _v∈ ex_κ(v) and summing over all κ:e→ℕκ:e turns it into an identity of symmetric functions: the left hand side becomes Xe=p1r−prX_e=p_1^r-p_r for the hypergraph consisting of the single hyperedge e, and the generating function of C(τ,B)C(τ,B) is FB(r)F_B^(r) for each of the r!r! orders τ, so p1r−pr=∑S⊆[r−1](Ar(S)−(−1)|S|)FS(r).p_1^r-p_r= _S [r-1] (A_r(S)-(-1)^|S| )F_S^(r). Applying equation 39 to an antichain on r elements gives p1r=∑π∈rFDes(π)(r)=∑SAr(S)FS(r)p_1^r= _π∈ S_rF^(r)_Des(π)= _SA_r(S)F_S^(r), so the display above is the classical expansion pr=∑S⊆[r−1](−1)|S|FS(r)p_r= _S [r-1](-1)^|S|F_S^(r) of the power sum in the fundamental basis. The content of Proposition C.14.4 is that the identity already holds coloring by coloring, which is what makes it glue, and the content of Lemma C.14.2 is that its coefficients are nonnegative. Lemma C.14.9. For every Ω one has KΩ=Γ(PΩ,ωΩ)K_ = (P_ , _ ). In particular KΩK_ is a nonnegative integral combination of fundamental quasisymmetric functions. Proof. Suppose first that κ satisfies equation 36. Then along every oriented edge u→vu→ v of TΩT_ one has κ(u)≤κ(v)κ(u)≤κ(v), with strict inequality when the edge is strict. If x<PΩyx<_P_ y there is a directed path from x to y in TΩT_ , and chaining the inequalities along it gives κ(x)≤κ(y)κ(x)≤κ(y). Suppose in addition that ωΩ(x)>ωΩ(y) _ (x)> _ (y). By equation 38 the value of ωΩ _ increases along every weak edge, so if all edges of that directed path were weak we would get ωΩ(x)<ωΩ(y) _ (x)< _ (y). Hence the path contains a strict edge, so at least one of the chained inequalities is strict and κ(x)<κ(y)κ(x)<κ(y). Thus κ is a (PΩ,ωΩ)(P_ , _ )-partition. Conversely, suppose κ is a (PΩ,ωΩ)(P_ , _ )-partition and let u→vu→ v be an oriented edge of TΩT_ , so that u<PΩvu<_P_ v. If the edge is weak then ωΩ(u)<ωΩ(v) _ (u)< _ (v) by equation 38 and the definition gives κ(u)≤κ(v)κ(u)≤κ(v). If it is strict then ωΩ(u)>ωΩ(v) _ (u)> _ (v) and the definition gives κ(u)<κ(v)κ(u)<κ(v). Ranging over the edges contributed by a hyperedge e recovers exactly the conditions equation 36 for e. The colorings counted by KΩK_ are therefore precisely the (PΩ,ωΩ)(P_ , _ )-partitions, and the last assertion follows from equation 39. ∎ C.14.4 Proof of the main theorem Proof of Theorem C.14.1. We may assume n≥2n≥ 2. Every weight WΩW_ in equation 37 is nonnegative by Lemma C.14.2, and every KΩK_ is a nonnegative integral combination of fundamental quasisymmetric functions by Lemma C.14.9. Hence equation 37 exhibits XHX_H as a nonnegative rational combination of the FS(n)F_S^(n), so every aSa_S is a nonnegative rational number. It remains to see that the aSa_S are integers. For T⊆[n−1]T [n-1] let α(T)=(α1(T),…,αm(T))α(T)=( _1(T),…, _m(T)) be the associated composition of n, defined as above with r replaced by n, and let MT(n)=∑i1<⋯<imxi1α1(T)⋯ximαm(T)M_T^(n)= _i_1<·s<i_mx_i_1 _1(T)·s x_i_m _m(T) be the corresponding monomial quasisymmetric function. Splitting the defining sum of FS(n)F_S^(n) according to the exact set T of indices j with ij<ij+1i_j<i_j+1 gives FS(n)=∑T⊇SMT(n).F_S^(n)= _T SM_T^(n). Write XH=∑TcTMT(n)X_H= _Tc_TM_T^(n). Then cT=∑S⊆TaSc_T= _S Ta_S, so Möbius inversion in the Boolean lattice gives aS=∑T⊆S(−1)|S|−|T|cT.a_S= _T S(-1)^|S|-|T|c_T. Now cTc_T is the coefficient of x1α1(T)⋯xmαm(T)x_1 _1(T)·s x_m _m(T) in XHX_H, which is the number of proper colorings κ:V→[m]κ:V→[m] with |κ−1(i)|=αi(T)|κ^-1(i)|= _i(T) for every i∈[m]i∈[m], hence a nonnegative integer. Therefore each aSa_S is an integer, and being also nonnegative it lies in ℤ≥0Z_≥ 0. ∎ Remark C.14.10. The weighted decomposition produces no splittings, and hence says nothing about the partitionability of the Coxeter complexes of type Ar−1A_r-1 with the empty face removed. Nor does it recover the combinatorial interpretation of the coefficients that Taylor 2015 gives in the prime case. Theorem C.14.1 establishes that the aSa_S are nonnegative integers, but the decomposition equation 37 realizes them only as nonnegative rational combinations of numbers of linear extensions of posets. Remark C.14.11. Connectedness enters the argument only in Lemma C.14.6 and in the proof of Lemma C.14.7, where it supplies a path of H between any two vertices. For a hyperforest, that is, a disjoint union of hypertrees, the same construction produces a spanning forest TΩT_ whose components are the trees built inside the components of H, and PΩP_ is the disjoint union of the corresponding posets; equation 39 applies verbatim and Theorem C.14.1 holds for hyperforests as well. References. Breuer & Klivans (2016) Felix Breuer and Caroline J Klivans. Scheduling problems. Journal of Combinatorial Theory, Series A, 139:59–79, 2016. Gessel (1984) Ira M Gessel. Multipartite P-partitions and inner products of skew schur functions. Contemporary Mathematics, 34:289–317, 1984. Gessel & Kalikow (2005) Ira M Gessel and Louis H Kalikow. Hypergraphs and a functional equation of bouwkamp and de bruijn. Journal of Combinatorial Theory, Series A, 110(2):275–289, 2005. Gessel & Reutenauer (1993) Ira M Gessel and Christophe Reutenauer. Counting permutations with given cycle structure and descent set. Journal of Combinatorial Theory, Series A, 64(2):189–215, 1993. Pawlowski (2018) Brendan Pawlowski. Chromatic symmetric functions via the group algebra of s_ns\_n. arXiv preprint arXiv:1802.05470, 2018. Stanley (1995) Richard P Stanley. A symmetric function generalization of the chromatic polynomial of a graph. Advances in Mathematics, 111(1):166–194, 1995. Stanley (1998) Richard P Stanley. Graph colorings and related symmetric functions: ideas and applications a description of results, interesting applications, & notable open problems. Discrete Mathematics, 193(1-3):267–286, 1998. Stanley (2011) Richard P Stanley. Enumerative combinatorics volume 1 second edition. Cambridge studies in advanced mathematics, 2011. Taylor (2015) Jair Taylor. Chromatic symmetric functions of hypertrees. arXiv preprint arXiv:1506.08262, 2015. C.15 Quartically many Fano subsquares in Latin squares This problem asks whether a Latin square of order n can contain more than cubically many subsquares of order 77. We show that it can: for every n≥56n≥ 56 there is a Latin square of order n with more than (n/28)4(n/28)^4 subsquares isotopic to the Fano square S7S_7. A quartic upper bound is already due to Browning et al. 2015, so it is the lower bound that settles the order of growth; we also prove the explicit upper bound n4n^4 by a short self-contained argument. The construction is an affine lift of the Fano quasigroup over a finite field of characteristic 22. The upper bound comes from the observation that three rows and one column already generate the whole incidence structure of S7S_7. C.15.1 Introduction Let M be a Latin square of order n. A subsquare of order m in M is a set of m rows together with a set of m columns whose induced m×m× m subarray contains only m distinct symbols; that subarray is then itself a Latin square, and its symbol set is determined by the chosen rows and columns. Two Latin squares are isotopic if one is carried to the other by relabeling rows, columns and symbols independently. Following Browning et al. 2014, write ζ(n,m)ζ(n,m) for the largest number of subsquares of order m in a Latin square of order n, and, for a fixed Latin square L, write ζ∗(n,L)ζ^*(n,L) for the largest number of subsquares isotopic to L in a Latin square of order n. Let V=23V=F_2^3 and P=V∖0P=V \0\. The Fano square S7S_7 is the Cayley table of the quasigroup (P,∘)(P, ) defined by x∘y=x,x=y,x+y,x≠y,x y= casesx,&x=y,\\ x+y,&x≠ y, cases (40) with addition in V. Equivalently, S7S_7 is the Steiner quasigroup of the Fano plane: the unordered triples x,y,x+y\x,y,x+y\ with x≠yx≠ y are exactly the seven lines of PG(2,2)PG(2,2). Writing 1,…,71,…,7 for the nonzero vectors of V in binary notation, S7S_7 is the array ∘123456711325476232167453213765445674123547615326745236176543217 array[]c|c &1&2&3&4&5&6&7\\ 1&1&3&2&5&4&7&6\\ 2&3&2&1&6&7&4&5\\ 3&2&1&3&7&6&5&4\\ 4&5&6&7&4&1&2&3\\ 5&4&7&6&1&5&3&2\\ 6&7&4&5&2&3&6&1\\ 7&6&5&4&3&2&1&7 array Wanless asked the following at the LOOPS ’11 open problem session (LOOPS ’11 2011, Problem 2.8). Fix a prime p. Is there a family of latin squares with more than cubically many subsquares of order p? More precisely, is it true that for every constant c there is a latin square L of order n such that there are more than cn3cn^3 subsquares of order p in L? The note accompanying the problem records that the answer is negative for p∈2,3,5p∈\2,3,5\, and that for p=7p=7 the subsquares would have to be multiplication tables of a Steiner quasigroup. Since the Steiner triple system of order 77 is unique up to isomorphism, the only Steiner quasigroup of order 77 is (P,∘)(P, ). The case p=7p=7 of the problem is therefore precisely the question, left open by Browning et al. 2014, of whether ζ∗(n,S7)ζ^*(n,S_7) grows faster than cubically. We answer it affirmatively. The results of Browning et al. 2014 give the exact order of growth of ζ(n,m)ζ(n,m) for several small m: one has ζ(n,m)=Θ(n3)ζ(n,m)= (n^3) for m∈2,3,5m∈\2,3,5\ and ζ(n,m)=Θ(n4)ζ(n,m)= (n^4) for m∈4,6,9,10m∈\4,6,9,10\, with the sharper bounds n3/8+O(n2)≤ζ(n,2)≤n3/4+O(n2)n^3/8+O(n^2)≤ζ(n,2)≤ n^3/4+O(n^2) and n3/27+O(n5/2)≤ζ(n,3)≤n3/18+O(n2)n^3/27+O(n^5/2)≤ζ(n,3)≤ n^3/18+O(n^2). For a fixed square they show that ζ∗(n,L)=Θ(n3)ζ^*(n,L)= (n^3) when L is cyclic, that ζ∗(n,L)=O(n3)ζ^*(n,L)=O(n^3) for a large class of L, and that every L admits some ε∈(0,1) ∈(0,1) with ζ∗(n,L)=Ω(n2+ε)ζ^*(n,L)= (n^2+ ). The value m=7m=7 is absent from both lists, and by the note quoted above the only order-77 isotopy class that can push ζ(n,7)ζ(n,7) past cubic growth is S7S_7. On the upper bound side, Browning et al. 2015 show that a Latin square of order n has O(nψ(m,t)+t)O(n^ψ(m,t)+t) subsquares of order m for all positive integers t≤m≤nt≤ m≤ n, where ψ(m,t)=⌈12⌊m/t⌋⌉ψ(m,t)= 12 m/t for odd m; with m=7m=7 and t=2t=2 this gives ζ(n,7)=O(n4)ζ(n,7)=O(n^4). For subsquares of unbounded order, Browning et al. 2013 prove that a Latin square of order n has at most nO(logk)n^O( k) subsquares of order k. Extremal behavior here is very far from typical behavior. McKay & Wanless 1999 show that for every ε>0 >0 almost all Latin squares of order n have at least n3/2−εn^3/2- subsquares of order 22, and Allsop & Wanless 2025 show that a uniformly random k×nk× n Latin rectangle has no proper subsquare of order 44 or more with probability 1−O(1/n)1-O(1/n). In particular a random Latin square of order n has, with probability 1−O(1/n)1-O(1/n), no proper subsquare of order 77, so a positive answer must come from an explicit construction. The main result of this section is the following. Theorem C.15.1. For every n≥56n≥ 56, (n28)4<ζ∗(n,S7)≤n4. ( n28 )^4<ζ^*(n,S_7)≤ n^4. In particular ζ∗(n,S7)=Θ(n4)ζ^*(n,S_7)= (n^4). Combined with the bound of Browning et al. 2015 quoted above, this settles the order of growth of ζ(n,7)ζ(n,7) as well, adding 77 to the list of m for which ζ(n,m)ζ(n,m) is known. Corollary C.15.2. ζ(n,7)=Θ(n4)ζ(n,7)= (n^4). We now summarize the construction. The key point is that S7S_7 is almost an 2F_2-linear object: off the diagonal the rule equation 40 is simply addition in V, and the diagonal rule x∘x=x x=x is the only obstruction to linearity. We therefore work on P×qP×F_q with q a power of two, keep the additive rule off the diagonal, and replace the diagonal rule by a nontrivial affine combination λu+(1+λ)vλ u+(1+λ)v in the fiber coordinate. The copies of S7S_7 are then the translated graphs of the 2F_2-linear maps V→qV _q: there are q3q^3 such maps and q translations, giving q4q^4 subsquares in a Latin square of order 7q7q. The affine parameter λ is what forces the diagonal cells of each graph to close up correctly. C.15.2 The construction Fix a power of two q≥4q≥ 4 and an element λ∈q∖0,1λ _q \0,1\. Define a binary operation ∗* on P×qP×F_q by (x,u)∗(y,v)=(x,λu+(1+λ)v),x=y,(x+y,u+v),x≠y,(x,u)*(y,v)= cases(x,\ λ u+(1+λ)v),&x=y,\\ (x+y,\ u+v),&x≠ y, cases (41) where the first coordinate is computed in V and the second in qF_q. Write LqL_q for the resulting array, with rows and columns indexed by P×qP×F_q. Lemma C.15.3. LqL_q is a Latin square of order 7q7q. Proof. Since |P×q|=7q|P×F_q|=7q, it suffices to check that ∗* is a quasigroup operation, that is, that for each row and each symbol there is a unique column producing that symbol, and likewise with the roles of rows and columns interchanged. Fix a row (x,u)(x,u) and a symbol (z,w)(z,w), and seek a column (y,v)(y,v) with (x,u)∗(y,v)=(z,w)(x,u)*(y,v)=(z,w). Suppose first that z=xz=x. The second branch of equation 41 would force x+y=z=x+y=z=x, hence y=0∉Py=0∉ P, so the first branch applies and y=xy=x. The remaining equation λu+(1+λ)v=wλ u+(1+λ)v=w has the unique solution v=(1+λ)−1(w+λu),v=(1+λ)^-1(w+λ u), because 1+λ≠01+λ≠ 0. Suppose next that z≠xz≠ x. The first branch would force z=xz=x, so the second branch applies and y=x+zy=x+z, which is nonzero and distinct from x because z≠xz≠ x and z≠0z≠ 0. The remaining equation u+v=wu+v=w has the unique solution v=w+uv=w+u. Now fix a column (y,v)(y,v) and a symbol (z,w)(z,w), and seek a row (x,u)(x,u). If z=yz=y then exactly as before x=yx=y, and λu+(1+λ)v=wλ u+(1+λ)v=w has the unique solution u=λ−1(w+(1+λ)v)u=λ^-1(w+(1+λ)v) because λ≠0λ≠ 0. If z≠yz≠ y then x=y+z∈Px=y+z∈ P with x≠yx≠ y, and u=w+vu=w+v. ∎ We next exhibit many Fano subsquares of LqL_q. Let ϕ:V→qφ V _q be an 2F_2-linear map and let r∈qr _q. Put r~=λ−1(1+λ)r, r=λ^-1(1+λ)r, and define Rϕ,r=(x,ϕ(x)+r):x∈P,Cϕ,r=(x,ϕ(x)+r~):x∈P, R_φ,r=\(x,φ(x)+r):x∈ P\, C_φ,r=\(x,φ(x)+ r):x∈ P\, Tϕ,r=(x,ϕ(x)+r+r~):x∈P. T_φ,r=\(x,φ(x)+r+ r):x∈ P\. Each of these sets has exactly seven elements, since its members have distinct first coordinates. Lemma C.15.4. For every pair (ϕ,r)(φ,r) the subarray of LqL_q on the rows Rϕ,rR_φ,r and the columns Cϕ,rC_φ,r is a subsquare with symbol set Tϕ,rT_φ,r, and it is isotopic to S7S_7. Proof. Index the row (x,ϕ(x)+r)(x,φ(x)+r), the column (y,ϕ(y)+r~)(y,φ(y)+ r) and the symbol (z,ϕ(z)+r+r~)(z,φ(z)+r+ r) by their first coordinates x,y,z∈Px,y,z∈ P. We claim that the row indexed by x times the column indexed by y is the symbol indexed by x∘yx y. Suppose x≠yx≠ y. The second branch of equation 41 applies, and since ϕφ is 2F_2-linear, (ϕ(x)+r)+(ϕ(y)+r~)=ϕ(x+y)+r+r~.(φ(x)+r)+(φ(y)+ r)=φ(x+y)+r+ r. The product is therefore the symbol indexed by x+y=x∘yx+y=x y. Suppose instead x=yx=y. The first branch of equation 41 applies, and in characteristic 22, λ(ϕ(x)+r)+(1+λ)(ϕ(x)+r~)=(λ+1+λ)ϕ(x)+λr+(1+λ)r~=ϕ(x)+λr+(1+λ)r~.λ (φ(x)+r )+(1+λ) (φ(x)+ r )=(λ+1+λ)φ(x)+λ r+(1+λ) r=φ(x)+λ r+(1+λ) r. The definition of r~ r says exactly that λr~=(1+λ)rλ r=(1+λ)r, and adding λr+r~λ r+ r to both sides of this identity turns it into λr+(1+λ)r~=r+r~.λ r+(1+λ) r=r+ r. The product is therefore the symbol indexed by x=x∘x=x x, which proves the claim. Consequently the subarray on Rϕ,r×Cϕ,rR_φ,r× C_φ,r uses exactly the seven symbols of Tϕ,rT_φ,r, so it is a subsquare of order 77, and the three indexings by elements of P exhibit an isotopism from S7S_7 onto it. ∎ Lemma C.15.5. The q4q^4 subsquares produced by Lemma C.15.4 are pairwise distinct. Proof. There are q3q^3 2F_2-linear maps V→qV _q and q choices of r, so there are q4q^4 pairs (ϕ,r)(φ,r); it suffices to show that distinct pairs give distinct row sets. Suppose Rϕ,r=Rψ,sR_φ,r=R_ψ,s. Each of the two sets contains exactly one element with first coordinate x, for every x∈Px∈ P, so ϕ(x)+r=ψ(x)+sφ(x)+r=ψ(x)+s for every x∈Px∈ P. Thus the 2F_2-linear map h=ϕ+ψh=φ+ψ takes the constant value r+sr+s on P. Choosing linearly independent a,b∈Va,b∈ V gives r+s=h(a+b)=h(a)+h(b)=(r+s)+(r+s)=0.r+s=h(a+b)=h(a)+h(b)=(r+s)+(r+s)=0. Hence r=sr=s, and h vanishes on P and at 00, so ϕ=ψφ=ψ. ∎ Corollary C.15.6. For every power of two q≥4q≥ 4 one has ζ∗(7q,S7)≥q4ζ^*(7q,S_7)≥ q^4. Moreover ζ∗(n,S7)>(n/28)4ζ^*(n,S_7)>(n/28)^4 for every n≥56n≥ 56. Proof. The first assertion is immediate from Lemmas C.15.3, C.15.4 and C.15.5. For the second, let n≥56n≥ 56 and let q be the largest power of two with 14q≤n14q≤ n. Then q≥4q≥ 4, and maximality gives 28q>n28q>n, so q>n/28q>n/28. By Evans’ embedding theorem, every partial Latin square of order N embeds in a Latin square of every order at least 2N2N (Evans 1960). Applying this to LqL_q, which is a Latin square of order 7q7q and in particular a partial Latin square of order 7q7q, produces a Latin square M of order n≥14qn≥ 14q whose subarray on the first 7q7q rows and columns is LqL_q. Every subsquare of LqL_q is a subsquare of M, so M contains at least q4>(n/28)4q^4>(n/28)^4 subsquares isotopic to S7S_7. ∎ Example C.15.7. Take q=4q=4, so that 4=0,1,α,α2F_4=\0,1,α,α^2\ with α3=1α^3=1, and λ∈α,α2λ∈\α,α^2\. Then L4L_4 is a Latin square of order 2828 containing at least 44=2564^4=256 subsquares isotopic to S7S_7. C.15.3 The upper bound It is convenient to view a Latin square M of order n as a tripartite incidence structure. Its vertices are the n rows, the n columns and the n symbols of M, and each of the n2n^2 cells contributes the triple consisting of its row, its column and the symbol it carries. The defining property of a Latin square is exactly that any two vertices lying in different parts belong to a unique common triple. Isotopisms are precisely the isomorphisms of these structures that respect the three parts. Observe that if K is a subsquare of M and two of the three vertices of some triple of M belong to K, then so does the third. Indeed, if a row and a column of K are given then the symbol in that cell is a symbol of K; if a row and a symbol of K are given then the column in which that symbol occurs in that row of K is the unique such column in M; and the third case is symmetric. For x∈Px∈ P write RxR_x, CxC_x and TxT_x for the row, column and symbol vertices of S7S_7 indexed by x, so that RxR_x, CyC_y and Tx∘yT_x y form a triple for all x,y∈Px,y∈ P. Lemma C.15.8. Let e1,e2,e3e_1,e_2,e_3 be a basis of V=23V=F_2^3 and put e4=e1+e2+e3e_4=e_1+e_2+e_3. Then the four vertices Re1,Re2,Re3,Ce4R_e_1,R_e_2,R_e_3,C_e_4 generate all twenty-one vertices of S7S_7 under the rule that two known vertices in different parts determine the third vertex of their triple. Proof. Throughout we use equation 40; note that ei≠e4e_i≠ e_4 for i∈1,2,3i∈\1,2,3\, since e4=eie_4=e_i would force the sum of the other two basis vectors to vanish. Pairing each of Re1,Re2,Re3R_e_1,R_e_2,R_e_3 with Ce4C_e_4 produces Te2+e3,Te1+e3,Te1+e2,T_e_2+e_3, T_e_1+e_3, T_e_1+e_2, because ei∘e4=ei+e4e_i e_4=e_i+e_4. Next, if x∈Px∈ P and x∘y=wx y=w with w≠xw≠ x, then y≠xy≠ x and y=x+wy=x+w is determined. Applying this three times, Re1,Te1+e3giveCe3,Re1,Te1+e2giveCe2,Re2,Te1+e2giveCe1.R_e_1,T_e_1+e_3\ give\ C_e_3, R_e_1,T_e_1+e_2\ give\ C_e_2, R_e_2,T_e_1+e_2\ give\ C_e_1. The pairs Rei,CeiR_e_i,C_e_i then give TeiT_e_i for i∈1,2,3i∈\1,2,3\, since ei∘ei=eie_i e_i=e_i. Applying the same rule as before, Re1,Te2giveCe1+e2,Re1,Te3giveCe1+e3,Re2,Te3giveCe2+e3.R_e_1,T_e_2\ give\ C_e_1+e_2, R_e_1,T_e_3\ give\ C_e_1+e_3, R_e_2,T_e_3\ give\ C_e_2+e_3. At this stage the columns Ce1,Ce2,Ce3,Ce1+e2,Ce1+e3,Ce2+e3,Ce4C_e_1,\ C_e_2,\ C_e_3,\ C_e_1+e_2,\ C_e_1+e_3,\ C_e_2+e_3,\ C_e_4 are all known, and these are all seven columns of S7S_7. Pairing Re1R_e_1 with each of them yields the symbols Te1∘yT_e_1 y for y∈Py∈ P; as y runs over P the element e1∘ye_1 y runs over all of P, so every symbol vertex is known. Pairing Ce1C_e_1 with each symbol vertex then yields every row vertex, since for each z∈Pz∈ P there is a unique x∈Px∈ P with x∘e1=zx e_1=z. ∎ Corollary C.15.9. For every n one has ζ∗(n,S7)≤n4ζ^*(n,S_7)≤ n^4. Proof. Let M be a Latin square of order n and let K be a subsquare of M isotopic to S7S_7, and let e1,e2,e3,e4e_1,e_2,e_3,e_4 be as in Lemma C.15.8. Choose an isotopism θ from S7S_7 onto K and record the quadruple (θ(Re1),θ(Re2),θ(Re3),θ(Ce4)), (θ(R_e_1),\ θ(R_e_2),\ θ(R_e_3),\ θ(C_e_4) ), which consists of three rows and one column of M. There are at most n4n^4 such quadruples, so it suffices to show that the quadruple determines K. Since θ is an isotopism onto K, it carries triples of S7S_7 to triples of M that lie in K. By the observation above, applying the rule “two vertices in different parts determine the third vertex of their triple” inside M to vertices of K never leaves K, and it agrees with the corresponding rule in S7S_7 transported by θ. By Lemma C.15.8, starting from the recorded quadruple and iterating this rule inside M therefore produces exactly the twenty-one vertices of K. Hence K is determined by the quadruple, as desired. ∎ C.15.4 Proof of the main theorem Proof of Theorem C.15.1. Corollary C.15.6 gives ζ∗(n,S7)>(n/28)4ζ^*(n,S_7)>(n/28)^4 for every n≥56n≥ 56, and Corollary C.15.9 gives ζ∗(n,S7)≤n4ζ^*(n,S_7)≤ n^4 for every n. ∎ Remark C.15.10. Along the sequence n=7qn=7q with q a power of two, Corollary C.15.6 gives the stronger bound ζ∗(n,S7)≥(n/7)4ζ^*(n,S_7)≥(n/7)^4, so that 12401≤lim supn→∞ζ∗(n,S7)n4≤1. 12401≤ _n→∞ ζ^*(n,S_7)n^4≤ 1. The correct constant remains undetermined. Proof of Corollary C.15.2. Every subsquare isotopic to S7S_7 has order 77, so ζ(n,7)≥ζ∗(n,S7)=Ω(n4)ζ(n,7)≥ζ^*(n,S_7)= (n^4) by Theorem C.15.1. In the other direction, the bound of Browning et al. 2015 with m=7m=7 and t=2t=2 gives ζ(n,7)=O(n4)ζ(n,7)=O(n^4). ∎ References. Allsop & Wanless (2025) Jack Allsop and Ian M Wanless. Subsquares in random latin rectangles. Combinatorica, 45(3):29, 2025. Browning et al. (2013) Joshua Browning, Douglas S Stones, and Ian M Wanless. 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