{"schema":"postcutoff/event@1","as_of":"2026-10-20T07:28:00+02:00","url":"https://postcutoff.com/e/2026-10-19-erdos-szemeredi-sunflower-conjecture-gpt-6-astra-pro/","md":"https://postcutoff.com/e/2026-10-19-erdos-szemeredi-sunflower-conjecture-gpt-6-astra-pro/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":"2026-10-19-erdos-szemeredi-sunflower-conjecture-gpt-6-astra-pro","date":"2026-10-19","date_precision":"day","short_title":"Erdős–Szemerédi sunflower conjecture proved for every k ≥ 3","deck":"The key vector bound (Theorem 2) was found by ChatGPT 6 Astra Pro (Bucić, He, Lee)","takeaway":"A 1970s Erdős–Szemerédi conjecture is claimed proved for every k: families of subsets of [n] with no k-sunflower have size at most c_k^n with c_k < 2. Before this only k = 3 was known, through the 2016 cap-set breakthrough. The authors say ChatGPT 6 Astra Pro found the proof of the central theorem. The paper also says a Lean check exists, but it has not been released yet.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"arXiv 2610.26686: Proof of the Erdős-Szemerédi sunflower conjecture (Bucić, He, Lee)","url":"https://arxiv.org/abs/2610.26686","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"Full text (PDF)","url":"https://arxiv.org/pdf/2610.26686","type":"paper","group":"primary","domain":"arxiv.org"}],"official":2,"filed":"2026-10-20","updated":"2026-10-20","orgs":["University of Vienna","KAIST","IBS","OpenAI"],"title":"Erdős–Szemerédi sunflower conjecture proved for every k ≥ 3; the key vector bound (Theorem 2) was found by ChatGPT 6 Astra Pro (Bucić, He, Lee)","summary":"On 19 Oct 2026 Matija Bucić, Fankang He (University of Vienna) and Hyunwoo Lee (KAIST/IBS) posted arXiv 2610.26686, \"Proof of the Erdős-Szemerédi sunflower conjecture\". Theorem 1: for every k ≥ 3 and n ≥ 1, a family of subsets of [n] with no k-sunflower has at most 2^((1 − 1/(16k³))·n) members. Until now only k = 3 was known, via Croot–Lev–Pach, Ellenberg–Gijswijt and the Alon–Shpilka–Umans reduction. The proof goes through a new bound on vector sunflowers (Theorem 2). Its \"Statement on the use of AI\" says: \"The proof of Theorem 2 was found by ChatGPT 6 Astra Pro\". The authors say they checked everything, that the proof \"has been verified in Lean 4.35\" and that the formalization \"will be made available on the Palomar repository\". It is an unrefereed preprint, and the Lean files were not public on 20 Oct.","key_facts":["Paper: 'Proof of the Erdős-Szemerédi sunflower conjecture', Matija Bucić, Fankang He (Faculty of Mathematics, University of Vienna), Hyunwoo Lee (KAIST and IBS ECOPRO), arXiv 2610.26686 [math.CO], submitted 19 Oct 2026","Conjecture (abstract): 'In the 1970s, Erdős and Szemerédi conjectured that for any n ≥ k − 1 ≥ 2, any collection of subsets of [n] not containing a k-sunflower has size at most c_k^n for some c_k < 2.' The paper traces the question back to 1970","Theorem 1: any subset of 2^[n] with no sunflower of k sets has size at most 2^((1 − 1/(16k³))n). Lower bound (Deuber, Erdős, Gunderson, Kostochka, Meyer): k-sunflower-free families of size 2^(n − O(n log log k / log k)), so the result is 'not far from best possible for large k'","Earlier: Erdős–Szemerédi 2^(n − Ω(√n)); Alweiss–Lovett–Wu–Zhang (2019) and Rao give 2^(n − Ω(n/log n)). k = 3 followed from the 2016 cap-set bounds (Croot–Lev–Pach; Ellenberg–Gijswijt) through Alon–Shpilka–Umans; Naslund and Sawin gave base 3/2^(2/3) ≈ 1.89 using Tao's slice rank","Method: Theorem 2 bounds the number V_k(D, n) of vectors in [D]^n without a k-sunflower by the polynomial method (dimension counting with low-degree factors and a Chernoff bound); a reduction (Section 4) transfers the vector bound to set families with D = 16k²","AI statement, verbatim: 'The proof of Theorem 2 was found by ChatGPT 6 Astra Pro, following a short discussion, which started by asking it to generalize the k = 3 case argument in [45], which we needed for our original target of proving the induced variant of the Erdős–Sauer problem. The original argument it produced used the slice-rank terminology due to Tao (as used in [45]); we rewrote it in the polynomial method language … The authors have verified all the results in the paper and take full responsibility for the content. The proof has been verified in Lean 4.35; the verification will be made available on the Palomar repository.' ([45] is Naslund–Sawin)","Concurrent work cited in the paper: another team proves that the set-sunflower relation on the Boolean cube has maximal slice rank, and 'They discuss extensive use of AI, including multi-agent setups'. The authors say that obstruction concerns set sunflowers, while their tester is for vector sunflowers, so it does not apply","Consequences listed in the paper: the vector bound extends the Alon–Shpilka–Umans question (connected to fast matrix multiplication) and gives results in arithmetic property testing and on equal subset sums","Reactions: no Hacker News thread or expert comment found by 20 Oct morning (web search; sweep HN list). A web search instead returned 2023 sources saying Naslund–Sawin had 'proved' the conjecture; that wording refers to the k = 3 case"],"key_numbers":[],"tags":["math","combinatorics","extremal-set-theory","sunflower","polynomial-method","slice-rank","cap-set","erdos","gpt-6-astra","lean"],"science":{"field":"mathematics","subfield":"extremal combinatorics","problem":"Erdős–Szemerédi sunflower conjecture: k-sunflower-free families in 2^[n] have size at most c_k^n with c_k < 2","result":"Claimed proof for every k ≥ 3, with the bound 2^((1 − 1/(16k³))n), via a new polynomial-method bound for vector sunflowers.","open_since":"1970","ai_system":["ChatGPT 6 Astra Pro"],"human_role":"AI-assisted: the authors posed the generalization of Naslund–Sawin and rewrote the argument; ChatGPT 6 Astra Pro found the proof of the key Theorem 2","verification":{"key":"lean","label":"Lean-verified","text":"Unrefereed arXiv preprint; authors checked all results; the stated Lean 4.35 proof is not public yet (promised for Palomar)"},"status":"pending","shock":"A conjecture open for about 50 years, solved before only for k = 3 by the cap-set breakthrough, fell after a 'short discussion' with a model.","short":"Erdős–Szemerédi sunflower conjecture, all k"},"body_md":"## What happened\n\nA sunflower is a family of sets whose pairwise intersections are all the same set. Erdős and Szemerédi asked how large a family\nof subsets of an n-element set can be without containing k sets forming a sunflower, and conjectured an exponential saving over\nthe trivial 2^n. The k = 3 case became a corollary of the 2016 cap-set breakthrough. Bucić, He and Lee had asked ChatGPT 6 Astra\nPro to generalize Naslund and Sawin's k = 3 slice-rank argument for a different problem, the induced Erdős–Sauer problem. The model\nproduced a bound for vector sunflowers that works for all k, and with the authors' reduction this gives the full conjecture.\n\nThe authors rewrote the model's slice-rank argument in polynomial-method language. They say all results were verified, including\nin Lean 4.35, but the formal proof had not been posted when the preprint appeared.\n\n## Why it matters\n\nThis is one of the best-known problems in extremal set theory. It sits next to the Erdős–Rado sunflower conjecture, which stays\nopen, and is tied to cap sets and the group-theoretic approach to matrix multiplication. The authors credit the model with the\nproof of the key theorem, not with exposition, which puts it among the strongest AI contributions to combinatorics so far. Until\nthe Lean files are released or experts review it, the claim stays pending.","disputed":[{"label":"Verification","text":"Unrefereed arXiv preprint; authors checked all results; the stated Lean 4.35 proof is not public yet (promised for Palomar)","sources":[]}],"related":[{"id":"2026-10-20-ai-assisted-results-october-20-listing","url":"https://postcutoff.com/e/2026-10-20-ai-assisted-results-october-20-listing/","date":"2026-10-20","date_precision":"day","short_title":"AI-assisted results in the 20 Oct 2026 arXiv listing","deck":"Poset dimension proofs found by GPT-6 Astra, Hitchin domination from ChatGPT 5.6 Sol's ideas, the section conjecture for generic curves, Yau's geometric conjecture, Erdős–Gyárfás r = 5, a Claude-built Gray-code reduction, and a Lean harness solving 6/6 IMO 2026","takeaway":"Tuesday's arXiv announcement was the strongest maths batch yet. Seven results have their own entries, among them the Erdős–Szemerédi sunflower conjecture, Erdős–Hajnal for all paths and a non-sofic hyperbolic group. This page catalogues the next tier: about forty papers whose AI disclosures run from 'the proofs were found by ChatGPT-6 Astra' to proofreading.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":23,"official":23,"filed":"2026-10-20","updated":"2026-10-20","orgs":["OpenAI","Anthropic","DeepSeek","AWS","various mathematicians"]},{"id":"2026-10-17-erdos-hajnal-all-paths-nguyen-astra-deepmind","url":"https://postcutoff.com/e/2026-10-17-erdos-hajnal-all-paths-nguyen-astra-deepmind/","date":"2026-10-17","date_precision":"day","short_title":"Erdős–Hajnal conjecture proved for every path P_t","deck":"Tung H. Nguyen, with ChatGPT-6 Astra proving key lemmas; a DeepMind internal model reached the same result independently, and Claude Opus 5.5 manuscripts claim P6 and all paths","takeaway":"Until now the Erdős–Hajnal property was known for paths only up to five vertices (P5, 2023). Oxford's Tung H. Nguyen claims it for every path, with GPT-6 Astra proving parts of the key lemmas. His paper says a Google DeepMind internal model got the P6 case and then all paths on its own, Lean-checked, and that two Claude Opus 5.5 manuscripts on GitHub claim P6 and all paths.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":4,"official":4,"filed":"2026-10-20","updated":"2026-10-20","orgs":["University of Oxford","OpenAI","Google DeepMind","Google Research","Anthropic"]},{"id":"2026-08-18-palomar-lean-registry","url":"https://postcutoff.com/e/2026-08-18-palomar-lean-registry/","date":"2026-08-18","date_precision":"day","short_title":"Palomar launches: a registry of Lean-verified mathematics to curb misrepresented AI proof claims","deck":null,"takeaway":"Formal verification became the main way to trust AI mathematics in 2026.","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":5,"official":5,"filed":"2026-09-29","updated":"2026-09-29","orgs":["Lean FRO","ICARM"]},{"id":"2026-05-27-sum-product-conjecture-false-over-reals","url":"https://postcutoff.com/e/2026-05-27-sum-product-conjecture-false-over-reals/","date":"2026-05-27","date_precision":"day","short_title":"Erdős–Szemerédi sum-product conjecture shown false over the reals","deck":"A GPT-5.5 Pro agent re-disproves it in 7 of 8 runs","takeaway":"It shows AI ideas spreading into human research and then being reproduced autonomously: a feedback loop between AI and human mathematicians.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":2,"official":2,"filed":"2026-09-29","updated":"2026-10-19","orgs":["OpenAI"]}],"people":[{"id":"terence-tao","name":"Terence Tao","url":"https://postcutoff.com/person/terence-tao/"}],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-20","type":"filed","text":"Created (PDF read in full text)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-20","run":null,"sources_read":"PDF read in full text","updated":"2026-10-20","human_review":null,"version":null},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":172,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":111,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":202,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":141,"in_training_data":false}],"short_url":"https://postcutoff.com/s/erdos-szemeredi-sunflower"}