{"schema":"postcutoff/event@1","as_of":"2026-10-19T09:28:00+02:00","url":"https://postcutoff.com/e/2026-10-16-max-hamiltonian-cycles-tournaments-gpt-5-6-wormald/","md":"https://postcutoff.com/e/2026-10-16-max-hamiltonian-cycles-tournaments-gpt-5-6-wormald/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":"2026-10-16-max-hamiltonian-cycles-tournaments-gpt-5-6-wormald","date":"2026-10-16","date_precision":"day","short_title":"Tournament Hamiltonian cycles: GPT-5.6 settles Alon's question","deck":null,"takeaway":"Princeton's Divoux and Wang asked GPT-5.6 for a lower bound; a second instance turned the method into the matching upper bound, closing a line of work that runs from Szele's 1943 paper through Alon (1990) and Friedgut–Kahn.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":[{"n":1,"title":"arXiv 2610.23755: The maximum number of Hamiltonian cycles in tournaments (Divoux, Wang)","url":"https://arxiv.org/abs/2610.23755","type":"paper","group":"primary","domain":"arxiv.org"}],"official":1,"filed":"2026-10-19","updated":"2026-10-19","orgs":["OpenAI","Princeton University"],"title":"Maximum number of Hamiltonian cycles in a tournament determined asymptotically, (cosh 1/cos 1)·(n−1)!/2ⁿ, answering Alon and proving Wormald's conjecture, with proofs 'originally found by GPT-5.6'","summary":"Alexander Divoux and Shouda Wang (Princeton) posted arXiv 2610.23755 on 16 Oct 2026. They prove that C(n), the maximum number of directed Hamiltonian cycles in an n-vertex tournament, is (cosh 1/cos 1 ± o(1))·(n−1)!/2ⁿ. The bound is attained asymptotically by the cyclic regular tournament on Z/nZ. This answers a question of Noga Alon, proves a conjecture of Nicholas Wormald, and also determines the maximum number of Hamiltonian paths asymptotically. \"We provide our own exposition of these proofs, which were originally found by GPT-5.6.\" GPT-5.6 first proved the lower bound, following the authors' hint to look for permanent estimates. A separate GPT-5.6 instance, given that proof, then generalized it to the matching upper bound.","key_facts":["Paper: Alexander Divoux and Shouda Wang, 'The maximum number of Hamiltonian cycles in tournaments', arXiv 2610.23755 [math.CO], submitted 16 Oct 2026, ~48 pp.","Main theorem: C(n) = (cosh 1 / cos 1 ± o(1))·(n−1)!/2ⁿ as n → ∞; extremal example: vertices Z/nZ, edge i→j iff j − i mod n < n/2","History (paper): Szele (1943) gave P(n) ≥ n!/2^(n−1), often called the first use of the probabilistic method, and an upper bound P(n) = O(n!·2^(−3n/4)); Alon (1990) C(n) = O(n^(3/2)(n−1)!/2ⁿ); Friedgut–Kahn O(n^α(n−1)!/2ⁿ), α ≈ 1.249; Adler–Alon–Ross C(n) ≥ (e − o(1))(n−1)!/2ⁿ; Wormald conjectured the regular cyclic tournament is extremal","The constant cosh 1/cos 1 was first guessed from numerics by Jay Pantone's computer program and communicated to Alon (acknowledgments)","AI role (acknowledgments): 'A proof was found by GPT-5.6 following some strategies supplied by the authors (the relevant one being a suggestion to search hard for suitable permanent estimates). In a surprising turn of events, after being input this proof and a few hopeful ideas, a separate instance of GPT-5.6 was able to generalize its techniques into an argument for the matching upper bound'","Human role: 'Many of the arguments were eventually replaced or simplified by the authors, and others originally came from the literature. The content of the present paper was written by the authors.' Noga Alon is thanked 'for many helpful discussions and comments'","Same pair (with Lowen) disproved Rota's 1970 matroid unimodality conjecture with ChatGPT 5.6 Pro in July (2026-07-24-rota-flats-unimodality-disproved)","No Hacker News thread or high-reach X post found on 19 Oct morning"],"key_numbers":[],"tags":["math","combinatorics","tournaments","hamiltonian-cycles","probabilistic-method","gpt-5-6","openai","ai-discovered-proof"],"science":{"field":"mathematics","subfield":"combinatorics (extremal and probabilistic; tournaments)","problem":"Maximum number of Hamiltonian cycles (and paths) in an n-vertex tournament: Alon's question (1990) and Wormald's conjecture on the extremal tournament","result":"C(n) = (cosh 1/cos 1 ± o(1))(n−1)!/2ⁿ, attained asymptotically by the cyclic regular tournament; asymptotic maximum number of Hamiltonian paths follows","open_since":"1943","ai_system":["GPT-5.6"],"human_role":"Authors chose the problem and the strategy hint (permanent estimates); GPT-5.6 found the lower-bound proof and a second instance the upper bound; authors rewrote, simplified and exposited","verification":{"key":"preprint","label":"Preprint only","text":"Unrefereed arXiv preprint; human-written exposition of the AI proofs"},"status":"pending","shock":"A second model instance turned a lower-bound proof into the matching upper bound, settling a classic extremal question that goes back to the first use of the probabilistic method."},"body_md":"## What happened\n\nA tournament is a complete graph with every edge given a direction. Counting directed Hamiltonian cycles in tournaments is one of the oldest problems in probabilistic combinatorics. Szele's 1943 bound is often called the first application of the probabilistic method, and the maximum has been pinned down only up to polynomial factors since Alon's 1990 paper. Numerics by Jay Pantone suggested the constant cosh 1/cos 1.\n\nDivoux and Wang first tried to prove the conjectured lower bound for the cyclic tournament, since it \"seemed promising for AI models to tackle\". GPT-5.6 did this with permanent estimates. A second GPT-5.6 instance, given the proof, extended the technique to the upper bound for every tournament, which settles the asymptotics. The paper is the authors' own exposition, and they replaced or simplified many of the model's arguments.\n\n## Why it matters\n\nThe result closes a classic extremal question from the start of the probabilistic method, and the authors say both directions of the proof came from the model. The upper bound, the harder direction, was a generalization the authors had not expected (\"in a surprising turn of events\").","disputed":[{"label":"Verification","text":"Unrefereed arXiv preprint; human-written exposition of the AI proofs","sources":[]}],"related":[{"id":"2026-10-19-ai-assisted-results-october-19-listing","url":"https://postcutoff.com/e/2026-10-19-ai-assisted-results-october-19-listing/","date":"2026-10-19","date_precision":"day","short_title":"AI-assisted results in the 19 Oct 2026 arXiv listing","deck":"Experts rework OpenAI's release papers (Deligne–Drinfeld, Hénon–Lane–Emden, Navier–Stokes), Wright's harmonic-analysis problems disproved with Codex, an Aaronson–Ambainis lower bound, and DeepMind's Busy Beaver agent","takeaway":"Monday's arXiv announcement brought the first wave of human follow-up papers on OpenAI's October maths release, alongside a dozen routine AI disclosures. The strongest items have their own entries: Brennan, Smith–Toda, lonely runner and tournament Hamiltonian cycles.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":14,"official":14,"filed":"2026-10-19","updated":"2026-10-19","orgs":["OpenAI","Anthropic","Google DeepMind","Harmonic","various mathematicians"]},{"id":"2026-07-24-rota-flats-unimodality-disproved","url":"https://postcutoff.com/e/2026-07-24-rota-flats-unimodality-disproved/","date":"2026-07-24","date_precision":"day","short_title":"Rota's 1970 unimodality conjecture for matroid flats disproved, with ChatGPT 5.6 Pro suggesting a key construction","deck":null,"takeaway":"Rota's conjecture was one of the best-known open problems about matroids.","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":4,"official":4,"filed":"2026-09-30","updated":"2026-09-30","orgs":["OpenAI","Anthropic"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-19","type":"filed","text":"Created (PDF read)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-19","run":null,"sources_read":"PDF read","updated":"2026-10-19","human_review":null,"version":{"date":"2026-10-19"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":169,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":108,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":199,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":138,"in_training_data":false}],"short_url":"https://postcutoff.com/s/tournament-hamiltonian-cycles"}