{"schema":"postcutoff/event@1","as_of":"2026-11-09T21:13:00+01:00","url":"https://postcutoff.com/e/2026-10-30-alon-krivelevich-conjecture-disproved-gpt-6-astra-xean/","md":"https://postcutoff.com/e/2026-10-30-alon-krivelevich-conjecture-disproved-gpt-6-astra-xean/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":"2026-10-30-alon-krivelevich-conjecture-disproved-gpt-6-astra-xean","date":"2026-10-30","date_precision":"day","short_title":"Alon–Krivelevich conjecture disproved with GPT-6 Astra","deck":null,"takeaway":null,"category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"arXiv 2610.42094: A Counterexample to the Alon–Krivelevich Conjecture (Junran Yu, Gexin Yu)","url":"https://arxiv.org/abs/2610.42094","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"Alon & Krivelevich (1998), The choice number of random bipartite graphs","url":"https://doi.org/10.1007/BF01608526","type":"paper","group":"primary","domain":"doi.org"}],"official":2,"filed":"2026-11-02","updated":"2026-11-02","orgs":["OpenAI","Nanjing Normal University","William & Mary"],"title":"Alon–Krivelevich conjecture on choosability of bipartite graphs disproved; counterexample and proof found by GPT-6 Astra (Junran Yu, Gexin Yu)","summary":"Junran Yu and Gexin Yu (arXiv 2610.42094, 30 Oct 2026) construct bipartite graphs with maximum degree Δ and choice number at least (1/32 − o(1))(ln Δ)²/(ln ln Δ)³, so ch(G)/ln Δ is unbounded, disproving the 1998 Alon–Krivelevich conjecture that bipartite graphs are O(log Δ)-choosable. The authors state that GPT-6 Astra, run through the \"xean\" research harness, found the counterexample and its proof, and that they verified everything. Unrefereed preprint.","key_facts":["Construction: for k ≥ 2 and r² > 2(2k−1) ln(2k−1), both vertex classes are copies of all k-subsets of a (2k−1)-set, adjacent when the sets meet in at most r elements; each vertex gets its own subset as list, and the graph is not colourable, so ch(G) ≥ k+1","Consequence: graphs with ch(G) ≥ (1/32 − o(1))(ln Δ)²/(ln ln Δ)³; the conjecture (Alon and Krivelevich, Annals of Combinatorics 1998) predicted ch = O(log Δ) for bipartite graphs","Tools: conditional product measures, nested coupling and Doob martingales written as weighted counting, plus Brouwer's fixed-point theorem","AI usage declaration (verbatim): 'We used the gpt-6-astra model through xean [4], a harness for AI-assisted mathematical research, to find the counterexample and its proof. Claude Opus 5.5 was used to rewrite the article. All mathematical content has been independently verified by the authors'","Verification status: preprint, author-verified; no Lean formalization stated; no outside review seen as of 2 Nov 2026"],"key_numbers":[],"tags":["math","combinatorics","graph-theory","list-coloring","gpt-6-astra","claude-opus-5-5","arxiv"],"science":{"field":"mathematics","subfield":"graph theory (list colouring)","problem":"Alon–Krivelevich conjecture: bipartite graphs of maximum degree Δ are O(log Δ)-choosable","result":"Disproved: bipartite graphs with ch(G) ≥ (1/32 − o(1))(ln Δ)²/(ln ln Δ)³","open_since":"1998","ai_system":["GPT-6 Astra","xean","Claude Opus 5.5"],"human_role":"AI-found: GPT-6 Astra found the counterexample and proof; authors verified it; Claude rewrote the text","verification":{"key":"preprint","label":"Preprint only","text":"Preprint, verified by the authors"},"status":"pending","shock":"A well-known list-colouring conjecture backed by Alon falls to a model-found explicit construction."},"body_md":"## What happened\n\nJunran Yu (Nanjing Normal University) and Gexin Yu (William & Mary) posted a counterexample to the Alon–Krivelevich conjecture. The bipartite graph has the k-subsets of a (2k−1)-set on both sides, and lists equal to the subsets themselves; the authors show no proper list colouring exists, giving a choice number that grows faster than ln Δ.\n\nTheir AI usage declaration says GPT-6 Astra, used \"through xean, a harness for AI-assisted mathematical research\", found \"the counterexample and its proof\", Claude Opus 5.5 rewrote the article, and the authors verified all content.\n\n## Why it matters\n\nThe conjecture, from Noga Alon and Michael Krivelevich's 1998 paper, was a central open question on list colouring of bipartite graphs (the best known upper bound is Δ/log Δ-type). If it holds up, it is one of the more prominent combinatorics conjectures refuted by a model-found construction. We did not see expert reactions yet.","disputed":[{"label":"Verification","text":"Preprint, verified by the authors","sources":[]}],"related":[],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-11-02","type":"filed","text":"Created from the 2 Nov sweep (arXiv AI-disclosure list)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-11-02","run":null,"sources_read":"The 2 Nov sweep (arXiv AI-disclosure list)","updated":"2026-11-02","human_review":null,"version":{"date":"2026-11-02"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":183,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":122,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":213,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":152,"in_training_data":false}],"short_url":"https://postcutoff.com/s/alon-krivelevich-conjecture"}