{"schema":"postcutoff/event@1","as_of":"2026-10-30T13:42:00+01:00","url":"https://postcutoff.com/e/2026-10-27-erdos-lovasz-intersecting-covering-number-frankl-ai-construction/","md":"https://postcutoff.com/e/2026-10-27-erdos-lovasz-intersecting-covering-number-frankl-ai-construction/index.md","disclosure":{"written_by":"AI agents (Claude Opus 5.5 in Claude Code)","editor":"human editor","policy":"https://postcutoff.com/about/"},"license":null,"id":"2026-10-27-erdos-lovasz-intersecting-covering-number-frankl-ai-construction","date":"2026-10-27","date_precision":"day","short_title":"Erdős–Lovász covering-number problem: AI-found construction disproves Frankl's 1996 conjecture","deck":null,"takeaway":"Erdős and Lovász showed in 1975 that an intersecting family of k-sets that needs k points to cover it has at most k^k members. The best constructions stayed near (k/2)^(k−1) for 30 years. In 1996 Frankl, Ota and Tokushige conjectured that m(k) < (ck)^k for some c < 1. A 27 Oct preprint by Peter Frankl, Jian Wang and two Sichuan University students builds families of size k^k·exp(−50√(k log k)). With Zakharov's 2024 upper bound this pins m(k) down to k^k·exp(−k^(1/2+o(1))) and refutes the 1996 conjecture. The authors write that 'the construction was found using generative AI tools'. They do not name the models.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"arXiv 2610.37315: The maximum size of intersecting k-graphs with covering number k (Frankl, Wang, Yang, Zhang)","url":"https://arxiv.org/abs/2610.37315","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"Zakharov, On the size of maximal intersecting families (CPC 2024; arXiv 2010.02541)","url":"https://arxiv.org/abs/2010.02541","type":"paper","group":"primary","domain":"arxiv.org"}],"official":2,"filed":"2026-10-29","updated":"2026-10-29","orgs":["Rényi Institute","Sichuan University"],"title":"Erdős–Lovász problem on intersecting k-graphs with covering number k nearly settled: a construction 'found using generative AI tools' gives m(k) = k^k·exp(−k^(1/2+o(1))) and disproves the 1996 Frankl–Ota–Tokushige conjecture, with Peter Frankl as co-author","summary":"\"The maximum size of intersecting k-graphs with covering number k\" (Peter Frankl, Jian Wang, Yaoran Yang, Yutong Zhang; arXiv 2610.37315, 27 Oct 2026) proves m(k) ≥ k^k·exp(−50√(k log k)) for log k ≥ 32, where m(k) is the largest intersecting k-uniform family with covering number k. Erdős and Lovász (1975) proved m(k) ≤ k^k. The best lower bound was (k/2)^(k−1) (Frankl–Ota–Tokushige, 1996), and the best upper bound is Zakharov's k^k·exp(−k^(1/2+o(1))) (2024). The two bounds now match up to the o(1) in the exponent, and the Frankl–Ota–Tokushige conjecture fails for large k. AI declaration: \"The construction was found using generative AI tools.\"","key_facts":["Main theorem (abstract): 'it is proved that for all k > e^32, m(k) ≥ k^k exp(−50√(k log k)). Together with Zakharov’s upper bound [12], this implies m(k) = k^k exp(−k^(1/2+o(1)))'","Conjecture refuted (Conjecture 1.4 in the paper, Frankl–Ota–Tokushige, JCTA 1996): 'There is an absolute constant c, with 1/2 < c < 1, such that m(k) < (ck)^k for all sufficiently large integers k.' The paper says Theorem 1.5 'disproves Conjecture 1.4 for all sufficiently large k'. Frankl, a co-author of that conjecture, is the first author","History as the paper recounts it: Erdős–Lovász (1975) proved m(k) ≤ k^k and m(k) ≥ ⌊k!(e − 1)⌋; Lovász conjectured the latter was exact; Frankl, Ota and Tokushige disproved that for k ≥ 4 with (k/2)^(k−1); upper bounds improved by Tuza, Arman–Retter, Frankl and Zakharov (CPC 2024). Small values: m(3) = 10 (Lovász), 42 ≤ m(4) ≤ 64","AI declaration (verbatim): 'The construction was found using generative AI tools. The authors verified the proofs and reworked the arguments to clarify the mathematical ideas underlying the construction and to provide a more conceptual exposition.' No model is named","Construction: a random intersecting k-graph built from an auxiliary hypergraph of classes of edges in edge-disjoint perfect matchings of K_2n, with many minimum covers (Theorem 1.6), then glued to a smaller Erdős–Lovász-type family","Status: unrefereed preprint, no formalization; no expert comment found as of 29 Oct"],"key_numbers":[],"tags":["math","combinatorics","extremal-set-theory","hypergraphs","erdos-lovasz","intersecting-families","generative-ai","arxiv"],"science":{"field":"mathematics","subfield":"extremal combinatorics (intersecting families, covers)","problem":"Erdős–Lovász problem: the maximum size m(k) of an intersecting k-uniform family with covering number k; Frankl–Ota–Tokushige conjecture m(k) < (ck)^k","result":"m(k) ≥ k^k·exp(−50√(k log k)) for log k ≥ 32, so m(k) = k^k·exp(−k^(1/2+o(1))) with Zakharov's bound; the Frankl–Ota–Tokushige conjecture is false for large k","open_since":"1975","ai_system":["unnamed generative AI tools"],"human_role":"AI found the construction; the authors verified the proofs, reworked the arguments and wrote the exposition","verification":{"key":"preprint","label":"Preprint only","text":"Unrefereed arXiv preprint"},"status":"pending","shock":"The lower bound jumps from about (k/2)^k to k^k up to a subexponential factor, and one of the conjecture's own authors signs the disproof of a construction found by AI."},"body_md":"## What happened\n\nA family of k-element sets is intersecting if any two members meet, and its covering number is the size of the smallest set that meets\nevery member. Since every member is itself such a set, the covering number of an intersecting family is at most k. Erdős and Lovász\nproved in 1975 that families reaching the maximum, τ = k, have at most k^k members. How large they can actually be has been studied\nsince. For thirty years the best constructions were near (k/2)^(k−1). In 1996 Frankl, Ota and Tokushige conjectured that the truth is\n(ck)^k with c < 1.\n\nThe new preprint gives a random construction of size k^k·exp(−50√(k log k)). Dmitrii Zakharov had shown in 2024 that m(k) is at most\nk^k·exp(−k^(1/2+o(1))), so the two bounds now agree up to the o(1) in the exponent. The authors write that the construction itself was\nfound with generative AI tools, and that they verified and rewrote the arguments.\n\n## Why it matters\n\nThe question was posed by Erdős and Lovász in a 1975 problem paper, and Peter Frankl is one of the best-known people working on\nintersecting families. The result settles the order of magnitude of m(k) and refutes a conjecture Frankl co-authored. It is another\ncase in which a model found a construction that experts had not expected. The paper does not name the models or say how much\nprompting was involved.","disputed":[{"label":"Verification","text":"Unrefereed arXiv preprint","sources":[]}],"related":[{"id":"2026-10-29-ai-assisted-results-october-29-listing-2","url":"https://postcutoff.com/e/2026-10-29-ai-assisted-results-october-29-listing-2/","date":"2026-10-29","date_precision":"day","short_title":"AI-assisted results, 29 Oct 2026 (part 2)","deck":"Claude 'largely carried out' Lipshitz's Khovanov research, ChatGPT found Soundararajan's proof and wrote Etingof's and Mathew's papers, George Elliott's free-product classification 'carried out using ChatGPT', and a first read of Hexagon's backlog","takeaway":"A second pass over the 28 Oct arXiv announcement and a first full read of Hexagon (hexagonmath.org), which the sweep had never covered, found results with large AI roles signed by senior mathematicians. Robert Lipshitz writes that his Khovanov paper's research 'was largely carried out by various Claude models'. Kannan Soundararajan's proof of a problem from Montgomery's CBMS book 'was found by ChatGPT 6'. ChatGPT 6 Pro wrote Pavel Etingof's answer to a 1999 question and produced Akhil Mathew's answer to a question of Drinfeld. George Elliott's classification of reduced free products of nuclear C*-algebras 'was carried out using ChatGPT'. Eight items have their own entries: the unforced 2D Euler non-uniqueness claims, the Erdős–Lovász covering-number bound, UPP^cc under intersection, Kisin's Serre conjecture, Linnik's constant ≤ 3.99, 3D Yang–Mills on a torus, random k-SAT and Chenglong Ma's Hovey–Strickland provenance claim.","category":"science","category_label":"Science & math","importance":2,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":30,"official":29,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI","Anthropic","Google DeepMind","various researchers"]},{"id":"2026-10-21-ai-assisted-results-october-21-listing","url":"https://postcutoff.com/e/2026-10-21-ai-assisted-results-october-21-listing/","date":"2026-10-21","date_precision":"day","short_title":"AI-assisted results in the 21 Oct 2026 arXiv listing","deck":"A ChatGPT 6.1 Sol counterexample to Frankl–Ota–Tokushige, klt Fano proofs from ChatGPT, Vince's conjecture with GPT Astra, Pólya for spherical shells, a Claude-built 55-addition 3×3 scheme and a Claude Opus 5.5 lemma in reverse mathematics","takeaway":"Wednesday's arXiv announcement brought four results with their own entries: the MDS conjecture in even characteristic, the double-cap conjecture, the quasitrace/monotone completeness counterexamples and the Neumann–Zagier conjecture. An expert verification of Claude's Hadamard matrices was added to that entry. This page lists the next tier, about thirty papers.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":18,"official":18,"filed":"2026-10-21","updated":"2026-10-21","orgs":["OpenAI","Anthropic","Google DeepMind","various mathematicians"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-29","type":"filed","text":"Created (arXiv PDF of 2610.37315)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-29","run":null,"sources_read":null,"updated":"2026-10-29","human_review":null,"version":{"date":"2026-10-29"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":180,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":119,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":210,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":149,"in_training_data":false}],"short_url":"https://postcutoff.com/s/erdos-lovasz-covering-number"}