{"schema":"postcutoff/event@1","as_of":"2026-10-30T13:42:00+01:00","url":"https://postcutoff.com/e/2026-10-28-murty-simon-conjecture-proof-gpt-5-6-sol-das-fox/","md":"https://postcutoff.com/e/2026-10-28-murty-simon-conjecture-proof-gpt-5-6-sol-das-fox/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":"2026-10-28-murty-simon-conjecture-proof-gpt-5-6-sol-das-fox","date":"2026-10-28","date_precision":"day","short_title":"Murty–Simon conjecture: proof for all n, bound found by GPT-5.6 Sol","deck":null,"takeaway":"The Murty–Simon conjecture (Plesník 1975, Murty and Simon) says a diameter-2-critical graph on n vertices has at most ⌊n²/4⌋ edges, with equality only for the balanced complete bipartite graph. Füredi proved it in 1992 only for n above a tower of 2s of height about 10¹⁴. On 28 Oct Sanjana Das and Jacob Fox (Stanford) posted a proof for every n. Their AI statement says the proof of the n²/4 bound 'was mostly found by AI, specifically GPT-5.6 Sol on August 11, 2026'. The authors simplified it and extended it to the equality case for odd n by hand. It is a preprint and has not been refereed.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":[{"n":1,"title":"arXiv 2610.39032: A proof of the Murty–Simon conjecture on diameter-2-critical graphs (Das, Fox)","url":"https://arxiv.org/abs/2610.39032","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"PDF with the AI statement","url":"https://arxiv.org/pdf/2610.39032","type":"paper","group":"primary","domain":"arxiv.org"}],"official":2,"filed":"2026-10-30","updated":"2026-10-30","orgs":["OpenAI","Stanford University"],"title":"Murty–Simon conjecture on diameter-2-critical graphs: Sanjana Das and Jacob Fox post a proof for all n, whose n²/4 bound was 'mostly found by AI' (GPT-5.6 Sol)","summary":"arXiv 2610.39032 (28 Oct 2026), \"A proof of the Murty–Simon conjecture on diameter-2-critical graphs\" by Sanjana Das and Jacob Fox, proves that every n-vertex diameter-2-critical graph has at most ⌊n²/4⌋ edges, with equality only for K_{⌈n/2⌉,⌊n/2⌋}, for every n. Before this, the conjecture was known only for astronomically large n through Füredi's 1992 regularity-lemma argument. GPT-5.6 Sol found a full proof of the bound and, for even n, of the equality case; the authors simplified the argument and proved the odd-n equality case themselves.","key_facts":["Claim (abstract): 'We present a proof of the Murty-Simon conjecture that works for all n. A proof of the ⌊n²/4⌋ bound was originally found by AI; we simplified this proof and extended its ideas to get a proof of the full conjecture.'","AI statement (verbatim): 'This proof was mostly found by AI, specifically GPT-5.6 Sol on August 11, 2026. It found a full proof of the n²/4 bound; we simplified some parts of this proof and cleaned up the presentation to get the argument in this paper. For even n, it also found a proof that the only equality case is K_{n/2,n/2}…' The odd-n equality case was done by the authors: 'We wrote up the argument on our own.' A final pass through GPT-6 Astra led to 'a few minor edits'","History (paper, §1): the statement goes back to Ore in the 1960s (per Erdős, via Füredi); Plesník conjectured the bound in 1975; Murty and Simon conjectured it independently, with the equality case. It would imply Mantel's theorem (1907), since every maximal triangle-free graph is diameter-2-critical. Bounds cn² with c approaching 1/4 came in the 1970s–80s","Füredi (1992) proved it for n greater than a tower of 2s of height roughly 10¹⁴, the first use of Szemerédi's regularity lemma to get an exact answer; the new proof starts from a maximal matching M of the graph and uses no regularity lemma","People: Jacob Fox (Stanford) is a leading extremal combinatorialist; Sanjana Das (Stanford). The AI found the proof in August, two and a half months before the preprint","Verification status: arXiv preprint, not refereed; no Lean formalisation mentioned; no HN thread or expert comment found on 30 Oct"],"key_numbers":[],"tags":["math","combinatorics","extremal-graph-theory","graph-theory","gpt-5-6-sol","gpt-6-astra","chatgpt","preprint"],"science":{"field":"mathematics","subfield":"extremal graph theory","problem":"Murty–Simon conjecture (Plesník 1975): diameter-2-critical graphs have at most ⌊n²/4⌋ edges, equality only for the balanced complete bipartite graph","result":"Claimed proof for every n (previously known only for n beyond a tower-type bound, Füredi 1992)","open_since":"1975","ai_system":["GPT-5.6 Sol","GPT-6 Astra"],"human_role":"AI-assisted: GPT-5.6 Sol found the n²/4 bound and the even-n equality case; the authors simplified it, proved the odd-n equality case and wrote the paper","verification":{"key":"preprint","label":"Preprint only","text":"Preprint by two established combinatorialists; not refereed, no formal proof"},"status":"pending","shock":"A problem whose only proof needed the regularity lemma and a tower-type threshold received an elementary proof for all n from a model run in August."},"body_md":"## What happened\n\nA graph has diameter 2 if any two vertices are joined by a path of length at most 2. It is diameter-2-critical if deleting any edge\nbreaks this. Maximal triangle-free graphs are examples, so a bound of n²/4 edges for all diameter-2-critical graphs would strengthen\nMantel's theorem. Plesník conjectured the bound in 1975, Murty and Simon added the equality case, and Füredi proved it in 1992 for\nenormous n only.\n\nOn 28 Oct Sanjana Das and Jacob Fox posted a proof for every n. Their AI statement says GPT-5.6 Sol found the proof of the bound and the\neven-n equality case on 11 August 2026. They simplified the argument, proved the odd-n equality case themselves and wrote the paper.\n\n## Why it matters\n\nThis is a well-known conjecture of extremal graph theory, and the AI role is stated plainly by a senior author. The proof avoids the\nregularity lemma, which was the only route known before. It still needs refereeing.","disputed":[{"label":"Verification","text":"Preprint by two established combinatorialists; not refereed, no formal proof","sources":[]}],"related":[{"id":"2026-10-30-ai-assisted-results-october-30-listing","url":"https://postcutoff.com/e/2026-10-30-ai-assisted-results-october-30-listing/","date":"2026-10-30","date_precision":"day","short_title":"AI-assisted results, 30 Oct: arXiv announcement of 29–30 Oct","deck":null,"takeaway":"The 30 Oct sweep listed 374 arXiv papers that mention AI. We skimmed all their AI statements and read about 35 PDFs. Six results got their own entries: Das–Fox's Murty–Simon proof (GPT-5.6 Sol found the bound), Colbrook–Stepaniants's spectral isometry proof (found by GPT-6 Astra, Lean), the Θ(log k log log k) ratio of greedy for online Steiner forest (GPT Astra, Lean), Gáspár's Laczkovich solution (key proposition by ChatGPT), Ren's pure-semisimplicity counterexample (GPT-6 Astra, Lean) and Petty's conjecture in dimension 5 (Tencent's hyprover agent). PNNL's OpenConjecture study also has one. This listing has the rest, among them a single-author ChatGPT 6 Pro proof of the Alon–Seymour–Thomas separator conjecture, b₄ ≤ 7 for Borsuk's problem and Fiedorowicz's conjecture from a Claude idea.","category":"science","category_label":"Science & math","importance":2,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":42,"official":42,"filed":"2026-10-30","updated":"2026-10-30","orgs":["OpenAI","Anthropic","Google","Tencent","various researchers"]},{"id":"2026-09-30-ai-assisted-conjecture-wave-summer-2026","url":"https://postcutoff.com/e/2026-09-30-ai-assisted-conjecture-wave-summer-2026/","date":"2026-09-30","date_precision":"day","short_title":"Summer 2026 flood: dozens of named conjectures settled on arXiv with disclosed AI help","deck":null,"takeaway":"This entry catalogues about 50 of them, with the AI role as the authors state it.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":11,"official":5,"filed":"2026-09-30","updated":"2026-10-19","orgs":["OpenAI","Anthropic","Google DeepMind","various mathematicians"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-30","type":"filed","text":"Created (science agent; arXiv abstract and PDF read, AI statement quoted; HN search found no thread)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-30","run":null,"sources_read":"Science agent; arXiv abstract and PDF read, AI statement quoted; HN search found no thread","updated":"2026-10-30","human_review":null,"version":{"date":"2026-10-30"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":181,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":120,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":211,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":150,"in_training_data":false}],"short_url":"https://postcutoff.com/s/murty-simon-conjecture-proof"}