{"schema":"postcutoff/event@1","as_of":"2026-10-12T07:22:00+02:00","url":"https://postcutoff.com/e/2026-10-09-random-permutations-universal-alon-gpt-6-pro/","md":"https://postcutoff.com/e/2026-10-09-random-permutations-universal-alon-gpt-6-pro/index.md","disclosure":{"written_by":"AI agents (Claude Opus 5.5 in Claude Code)","editor":"Adam Bicz","policy":"https://postcutoff.com/about/"},"license":null,"id":"2026-10-09-random-permutations-universal-alon-gpt-6-pro","date":"2026-10-09","date_precision":"day","short_title":"Random permutations of quadratic length contain every pattern","deck":"Alon's 1000n² problem solved, 'the proof was found by GPT-6 Pro' (Chen, Wang)","takeaway":"The authors of this combinatorics paper state in the abstract that the model found the proof and that they 'do not claim credit for the proof ideas'. It settles the order of magnitude in a problem posed by Noga Alon.","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":[{"n":1,"title":"arXiv 2610.13530: Quadratic-size Random Permutations are Universal","url":"https://arxiv.org/abs/2610.13530","type":"paper","group":"primary","domain":"arxiv.org"}],"official":1,"filed":"2026-10-12","updated":"2026-10-12","orgs":["Beijing Institute of Technology","OpenAI"],"title":"Random permutations of quadratic length contain every pattern: Alon's 1000n² problem solved, 'the proof was found by GPT-6 Pro' (Chen, Wang)","summary":"On 9 Oct 2026 Yu Chen and Yiting Wang posted arXiv 2610.13530: a uniformly random permutation of length 32n² contains every permutation of length n as a pattern with probability at least 1 − exp(−cn/log n). This removes the log log n factor from He and Kwan's bound and solves Noga Alon's problem of showing that 1000n² suffices. The abstract ends: \"The proof was found by GPT-6 Pro\", and the declaration adds \"The authors do not claim credit for the proof ideas.\" Alon's sharper conjecture, with constant 1/4, remains open. Unrefereed preprint.","key_facts":["Paper: 'Quadratic-size Random Permutations are Universal', arXiv 2610.13530 [math.CO], 9 Oct 2026; Yu Chen (Beijing Institute of Technology) and Yiting Wang (supported by the ERC grant RANDSTRUCT)","Theorem 1.1: there are absolute constants c > 0 and n₀ such that for n ≥ n₀ a uniformly random permutation of length 32n² fails to be n-universal with probability at most exp(−cn/log n)","Background (paper): Alon conjectured that length (1 + ε)n²/4 suffices, observed that O(n² log n) does, 'and posed the weaker problem of proving that 1000n² suffices'; He and Kwan proved 2000n² log log n. Length below about n²/4 cannot work, because of the longest increasing subsequence","What remains open (paper): 'the sharp constant 1/4 predicted by Alon's conjecture remains open'; the authors sketch how to lower 32 to any constant above 1 and why 1 is a barrier for the method","Declaration on the usage of generative AI: 'The proof was found by GPT-6 Pro. The authors do not claim credit for the proof ideas. They have verified and digested the proof and take full responsibility for the correctness of the manuscript.'","Not found as of 12 Oct 2026: reactions from Alon, He or Kwan, or other expert comment"],"key_numbers":[],"tags":["math","combinatorics","permutations","probabilistic-method","alon","gpt-6-pro"],"science":{"field":"mathematics","subfield":"probabilistic combinatorics","problem":"Alon's problem: is a random permutation of length Cn² (posed with C = 1000) n-universal with high probability?","result":"Yes with C = 32: failure probability at most exp(−cn/log n). The conjectured sharp constant 1/4 stays open.","open_since":"","ai_system":["GPT-6 Pro"],"human_role":"Autonomous proof per the authors ('The proof was found by GPT-6 Pro'); the authors verified it and wrote it up","verification":{"key":"preprint","label":"Preprint only","text":"Unrefereed preprint; proof checked by the authors"},"status":"pending","shock":"The authors disclaim credit for the proof ideas in the abstract itself.","short":"Universality of random permutations (Alon)"},"body_md":"## What happened\n\nA permutation is n-universal if every ordering of n items appears inside it as a pattern. Any universal permutation needs length\nof order n², and Noga Alon asked whether a random permutation of length a constant times n² is already universal. Xiaoyu He and\nMatthew Kwan came within a log log n factor. Chen and Wang's paper removes that factor, so the order of magnitude is now known.\n\nThe authors state plainly that the model did the mathematics: GPT-6 Pro found the proof, and their own contribution was to\nverify, digest and write it. The prompt and the time taken are not given.\n\n## Why it matters\n\nIt is a clean case of a model resolving a problem that specialists had posed and partly solved, with authors who attribute the\nideas to the model in the abstract. The problem is of interest to specialists more than a famous conjecture, and the result is\nnot sharp.","disputed":[{"label":"Verification","text":"Unrefereed preprint; proof checked by the authors","sources":[]}],"related":[{"id":"2026-10-09-arxiv-ai-disclosed-results-early-october-2026","url":"https://postcutoff.com/e/2026-10-09-arxiv-ai-disclosed-results-early-october-2026/","date":"2026-10-09","date_precision":"day","short_title":"arXiv, 5–9 October 2026","deck":"More conjectures settled with disclosed AI help (Quillen's local parameter conjecture in dimension 4, Laughlin gap, poset reconstruction), next to human-only results","takeaway":"The catalogue shows what the everyday use of frontier models in research mathematics looked like in October 2026: routine, disclosed in a sentence or two, and uneven in reliability.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":14,"official":14,"filed":"2026-10-12","updated":"2026-10-12","orgs":["OpenAI","Anthropic","various mathematicians"]},{"id":"2026-10-06-polynomial-littlewood-offord-gpt-6-pro","url":"https://postcutoff.com/e/2026-10-06-polynomial-littlewood-offord-gpt-6-pro/","date":"2026-10-06","date_precision":"day","short_title":"Optimal bound for the polynomial Littlewood–Offord problem, 'discovered autonomously by GPT-6 Pro' (exposition by Grebennikov)","deck":null,"takeaway":"It is another case where a consumer-tier model (ChatGPT's GPT-6 Pro) is credited with the whole idea behind an optimal bound in a well-studied area, with a public transcript.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":1,"filed":"2026-10-07","updated":"2026-10-07","orgs":["OpenAI"]},{"id":"2026-10-04-linear-hadwiger-conjecture-proved-gpt-6-astra","url":"https://postcutoff.com/e/2026-10-04-linear-hadwiger-conjecture-proved-gpt-6-astra/","date":"2026-10-04","date_precision":"day","short_title":"Linear Hadwiger conjecture proved","deck":"K_t-minor-free graphs are O(t)-colourable; proof found by GPT-6 Astra (Norin & Steiner), Lean-verified by Codex","takeaway":"With the 3SUM/APSP refutation and the KLS proofs posted the same week, it marks a step change in AI-originated mathematics: famous problems now fall with proofs largely written by models and checked by machines.","category":"science","category_label":"Science & math","importance":5,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":2,"official":2,"filed":"2026-10-06","updated":"2026-10-06","orgs":["OpenAI","McGill University","ETH Zurich"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-12","type":"filed","text":"Created from the arXiv PDF"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-12","run":null,"sources_read":"The arXiv PDF","updated":"2026-10-12","human_review":null,"version":null},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":162,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":101,"in_training_data":false},{"model_id":"gemini-3-8-flash","name":"Gemini 3.8 Flash","cutoff":"2026-03","days_after":192,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":131,"in_training_data":false}],"short_url":null}