{"schema":"postcutoff/event@1","as_of":"2026-10-30T13:42:00+01:00","url":"https://postcutoff.com/e/2026-10-06-openai-hadwiger-conjecture-counterexample-claim/","md":"https://postcutoff.com/e/2026-10-06-openai-hadwiger-conjecture-counterexample-claim/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":"2026-10-06-openai-hadwiger-conjecture-counterexample-claim","date":"2026-10-06","date_precision":"day","short_title":"Hadwiger's conjecture disproved? (OpenAI, Lean)","deck":null,"takeaway":"A 104-page manuscript credited to 'OpenAI' (dated 23 Sept, public on 6 Oct) claims arbitrarily large graphs whose chromatic number exceeds their largest clique minor. That would refute Hadwiger's 1943 conjecture, a far-reaching generalization of the four colour theorem known to hold only up to K6. The examples have no three pairwise non-adjacent vertices and also beat the fractional weakening. A Lean formalization of the ordinary counterexample, added in OpenAI's 7 Oct update, was accepted in a community comparator run, and an independent Lean project derived the fractional version. The graphs come from a random construction; no explicit example has been published, and no expert has published an assessment.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"OpenAI (PDF): A counterexample to Hadwiger's conjecture (23 Sept 2026)","url":"https://github.com/openai/math/blob/main/preprints/A-counterexample-to-Hadwigers-conjecture-September-23-2026/paper.pdf","type":"paper","group":"primary","domain":"github.com"},{"n":2,"title":"OpenAI (PDF): A counterexample to the Colin de Verdière chromatic conjecture","url":"https://github.com/openai/math/blob/main/preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/paper.pdf","type":"paper","group":"primary","domain":"github.com"},{"n":3,"title":"OpenAI (PDF): A linear list-coloring bound in terms of the Hadwiger number","url":"https://github.com/openai/math/blob/main/preprints/A-linear-list-coloring-bound-in-terms-of-the-Hadwiger-number-September-23-2026/paper.pdf","type":"paper","group":"primary","domain":"github.com"},{"n":4,"title":"OpenAI Lean scope note for family 157","url":"https://github.com/openai/math/blob/main/lean/docs/157.md","type":"code","group":"primary","domain":"github.com"},{"n":5,"title":"oai-math-revised comparator results (HadwigerCounterexample, ListHadwiger accepted)","url":"https://github.com/mathcommunity/oai-math-revised/blob/main/checks/comparator-2026-10-26/results.tsv","type":"code","group":"primary","domain":"github.com"},{"n":6,"title":"GitHub: EdouardBonnet/hadwiger-counterexample (Lean, fractional consequence)","url":"https://github.com/EdouardBonnet/hadwiger-counterexample","type":"code","group":"primary","domain":"github.com"},{"n":7,"title":"arXiv 2610.01946: Gu & Xu, A shorter proof of the linear list Hadwiger conjecture","url":"https://arxiv.org/abs/2610.01946","type":"paper","group":"primary","domain":"arxiv.org"},{"n":8,"title":"Wikipedia: Hadwiger conjecture (graph theory)","url":"https://en.wikipedia.org/wiki/Hadwiger_conjecture_(graph_theory)","type":"docs","group":"primary","domain":"en.wikipedia.org"},{"n":9,"title":"arXiv 2512.17114: Costa, Luu, Wood & Yip, Verifying Hadwiger's Conjecture for Examples of Graphs with α(G) = 2 (the test case)","url":"https://arxiv.org/abs/2512.17114","type":"paper","group":"primary","domain":"arxiv.org"},{"n":10,"title":"arXiv 2512.20392: Kühn, Sauermann, Steiner & Wigderson, Disproof of the Odd Hadwiger Conjecture","url":"https://arxiv.org/abs/2512.20392","type":"paper","group":"primary","domain":"arxiv.org"},{"n":11,"title":"Carlos Toledo: OpenAI's math release, what it claims beyond what it checks (finding G10)","url":"https://carlostoledo.co/reports/openai-math.html","type":"discussion","group":"reactions","domain":"carlostoledo.co"},{"n":12,"title":"Gil Kalai: Updates: Sharing AI progress on mathematics (amazing!) (7 Oct)","url":"https://gilkalai.wordpress.com/2026/10/07/updates-sharing-ai-progress-on-mathematics-amazing-and-my-lecture-plans/","type":"discussion","group":"reactions","domain":"gilkalai.wordpress.com"},{"n":13,"title":"oai-math-revised AGENT_NOTES for the Hadwiger manuscript","url":"https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/A-counterexample-to-Hadwigers-conjecture-September-23-2026/AGENT_NOTES.md","type":"discussion","group":"reactions","domain":"github.com"}],"official":10,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"],"title":"OpenAI's math release claims counterexamples to Hadwiger's conjecture (1943), even in its fractional form, using graphs with independence number at most two; the ordinary disproof is formalized in Lean","summary":"\"A counterexample to Hadwiger's conjecture\" (OpenAI, 23 Sept 2026, 104 pages; family 157 of the 6 Oct release) states: \"We disprove Hadwiger's conjecture by constructing arbitrarily large graphs whose chromatic number exceeds their Hadwiger number. The examples have independence number at most two, and even their ordinary fractional chromatic number exceeds their Hadwiger number.\" For m-vertex examples it proves h(G) < 26m/75 + 2/3 < m/2 ≤ χ_f(G) ≤ χ(G). Companion manuscripts in the family disprove the Colin de Verdière chromatic conjecture and prove the linear list-Hadwiger bound χ_list(G) ≤ C·h(G). A Lean formalization of the ordinary counterexample was added in OpenAI's 7 Oct update and accepted in the oai-math-revised comparator run (25–26 Oct); the fractional and Colin de Verdière statements are outside it. The construction is random and no explicit graph has been published.","key_facts":["Manuscript (abstract, verbatim): 'We disprove Hadwiger's conjecture by constructing arbitrarily large graphs whose chromatic number exceeds their Hadwiger number. The examples have independence number at most two, and even their ordinary fractional chromatic number exceeds their Hadwiger number. Thus they also disprove the fractional-coloring weakening discussed by Reed and Seymour.' 104 pages, dated 23 Sept 2026","Mechanism (paper, Theorem 1.1): arbitrarily large m-vertex graphs with α(G) ≤ 2 and connected matching number below m/100; a counting inequality gives h(G) ≤ (m + 4·cm(G) + 2)/3, while α(G) ≤ 2 forces χ_f(G) ≥ m/2. Corollary 1.2: h(G) < 26m/75 + 2/3 < m/2 ≤ χ_f(G) ≤ χ(G). By these bounds the chromatic number exceeds the Hadwiger number by a factor tending to 75/52 ≈ 1.44","Background: Hadwiger conjectured in 1943 that every graph with no K_t minor is (t−1)-colourable; Bollobás, Catlin and Erdős called it 'one of the deepest unsolved problems in graph theory'. It was known for t ≤ 6 (Robertson–Seymour–Thomas, 1993) and open for all larger t (Wikipedia). Its odd-minor strengthening was disproved by Kühn, Sauermann, Steiner and Wigderson (arXiv 2512.20392, Dec 2025), and the linear relaxation χ ≤ C·t was proved on 4 Oct 2026 by Norin and Steiner with a proof found by GPT-6 Astra (see 2026-10-04-linear-hadwiger-conjecture-proved-gpt-6-astra)","Companions in family 157 (all dated 23 Sept): 'A counterexample to the Colin de Verdière chromatic conjecture' (131 pages; χ exceeds the Colin de Verdière invariant by more than one, again with α ≤ 2) and 'A linear list-coloring bound in terms of the Hadwiger number' (42 pages; 'This resolves the Linear List Hadwiger conjecture affirmatively'). Gu and Xu (arXiv 2610.01946v2) say a key new ingredient of OpenAI's list-colouring proof is a linear choice-number bound for small K_t-minor-free graphs, and give a shorter proof of a weaker form of it","Lean: OpenAI's initial 6 Oct release formalized only the list-colouring bound (ListHadwiger). The challenge HadwigerCounterexample was added in the 7 Oct update (commit 3014888, pushed 8 Oct 05:03 UTC); the scope note says it proves h(G) < 26m/75 + 2/3 < m/2 ≤ χ(G) for arbitrarily large graphs with α ≤ 2, which 'disproves the ordinary chromatic form', with the fractional and Colin de Verdière statements 'outside this selected statement'. Both challenges were 'accepted' in the oai-math-revised comparator run of 25–26 Oct","Independent formal work: a Lean repository by the GitHub account EdouardBonnet vendors the release's construction and also derives the fractional consequence h(G) < 26m/75 + 2/3 < m/2 ≤ χ_f(G) ≤ χ(G); its README reports a complete local kernel replay on 11 Oct (Lax archive submission lax-342547)","No explicit witness: Carlos Toledo's audit lists the Hadwiger graphs among 15 headline objects that are 'random or sufficiently large constructions given only by a prescription', adding: 'Where the Lean is kernel-checked (Hadwiger, Sidorenko, Ryser) the existence is formally proved; the gap is that nobody can look at the object.' An AI-assisted survey (Ying Xie, arXiv 2610.17090) rates the method as new and notes the argument gives no explicit graph","Reactions: Gil Kalai (7 Oct): 'The new announcement gives counterexamples to the original conjecture, even among graphs with independence number at most two.' Wikipedia now says 'In October 2026, a bulk release by OpenAI included a claimed disproof of the conjecture.' No assessment by a structural graph theorist found (web, Hacker News, oai-math-revised notes) as of 29 Oct"],"key_numbers":[],"tags":["math","graph-theory","graph-coloring","graph-minors","hadwiger","counterexample","openai-math-release","lean","formal-verification"],"science":{"field":"mathematics","subfield":"graph theory (colouring and graph minors)","problem":"Hadwiger's conjecture (1943): every graph with no K_t minor is (t−1)-colourable, i.e. χ(G) ≤ h(G); also its fractional weakening χ_f(G) ≤ h(G)","result":"Claimed arbitrarily large counterexamples with independence number at most two, beating even the fractional form (Lean covers the ordinary form; an independent Lean project derives the fractional one); companion claims: Colin de Verdière chromatic conjecture false, linear list-Hadwiger bound true","open_since":"1943","ai_system":["OpenAI internal model (math release)"],"human_role":"Released as model output under OpenAI's fixed procedure, with no human author named; independent Lean re-packaging by a volunteer; no human exposition yet","verification":{"key":"peer-review","label":"Peer-reviewed","text":"Lean formalization of the ordinary counterexample (added 7 Oct), accepted by an independent comparator run; random construction with no published explicit graph; manuscript unrefereed"},"status":"pending","shock":"One of the central conjectures of graph theory is claimed false by a margin of about 44 percent, in dense graphs with no three mutually non-adjacent vertices, a class long studied as a test case."},"body_md":"## What happened\n\nHadwiger's conjecture says that a graph that cannot be contracted to a complete graph on t vertices can be coloured with t − 1\ncolours. For t = 5 it implies the four colour theorem. It was proved up to t = 6 in 1993 and stayed open beyond. Graphs in which no\nthree vertices are pairwise non-adjacent (independence number at most two) were a well-studied test case, because there colouring needs\nat least half as many colours as there are vertices.\n\nOpenAI's 6 Oct release claims counterexamples in exactly that class. The graphs are built at random so that they have no large\n\"connected matching\", which caps the size of any clique minor at about 35 percent of the vertices, while they still need at least 50\npercent as many colours. The Lean proof of this ordinary counterexample was not in the first release. OpenAI added it the next day,\nand a community comparator run later accepted it. A volunteer's separate Lean project derives the fractional version as well.\n\n## Why it matters\n\nIf the statement holds, one of the best-known conjectures in graph theory is false, and false by a constant factor, not by one colour.\nBecause the example is random and unpublished, nobody can inspect a concrete graph. The evidence is the kernel-checked existence proof\nplus the 104-page argument, which no graph theorist has publicly assessed. The same family also claims the opposite, positive\ndirection for list colouring, which complements Norin and Steiner's AI-found proof of the linear Hadwiger conjecture two days earlier.","disputed":[{"label":"Verification","text":"Lean formalization of the ordinary counterexample (added 7 Oct), accepted by an independent comparator run; random construction with no published explicit graph; manuscript unrefereed","sources":[]}],"related":[{"id":"2026-10-24-oai-math-revised-community-hub","url":"https://postcutoff.com/e/2026-10-24-oai-math-revised-community-hub/","date":"2026-10-24","date_precision":"day","short_title":"oai-math-revised: mathematicians launch a community hub to revise OpenAI's AI-written math manuscripts (guest post on Tao's blog)","deck":null,"takeaway":"This is the first organized attempt by research mathematicians to digest and credit the OpenAI catalogue.","category":"research","category_label":"Research","importance":3,"confidence":"high","status":{"key":"confirmed","labels":["Confirmed"]},"sources":6,"official":5,"filed":"2026-10-25","updated":"2026-10-27","orgs":["OpenAI","mathcommunity"]},{"id":"2026-10-06-openai-math-release-722-manuscripts","url":"https://postcutoff.com/e/2026-10-06-openai-math-release-722-manuscripts/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI releases 722 AI-written math manuscripts claiming hundreds of open problems","deck":"Including quasi-Riemann, Unique Games, Hodge for CM abelian varieties and free group factors","takeaway":"If even a fraction of these results hold up, this is the largest single jump in mathematical knowledge on record, produced by an AI system in about six weeks.","category":"science","category_label":"Science & math","importance":5,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":79,"official":13,"filed":"2026-10-07","updated":"2026-10-29","orgs":["OpenAI"]},{"id":"2026-10-06-openai-plane-not-five-colourable-hadwiger-nelson-claim","url":"https://postcutoff.com/e/2026-10-06-openai-plane-not-five-colourable-hadwiger-nelson-claim/","date":"2026-10-06","date_precision":"day","short_title":"Chromatic number of the plane is 6 or 7, OpenAI claims (Lean)","deck":null,"takeaway":"A 62-page manuscript credited to 'OpenAI' (dated 23 Sept, public on 6 Oct) claims that every 5-colouring of the plane has two points at distance one with the same colour, with no regularity assumed on the colour classes. That raises de Grey's 2018 lower bound of five to six, so the chromatic number of the plane is 6 or 7. The proof first transfers arbitrary colourings to a weak measurability condition, then rules out five colours under it. It is formalized in Lean and passed a community comparator run and a stricter independent check. Mathematicians called the method unlike anything in the prior literature; by compactness a finite non-5-colourable unit-distance graph must exist, but none has been displayed.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":10,"official":5,"filed":"2026-10-29","updated":"2026-10-29","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":"2026-10-19","orgs":["OpenAI","McGill University","ETH Zurich"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-29","type":"filed","text":"Created (CONTENTS entry, the three PDFs (104, 131, 42 pp), Lean scope note and its commit history, comparator results, Bonnet's repository, Kalai, Gu–Xu, oai-math-revised notes and Wikipedia read)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-29","run":null,"sources_read":"CONTENTS entry, the three PDFs (104, 131, 42 pp), Lean scope note and its commit history, comparator results, Bonnet's repository, Kalai, Gu–Xu, oai-math-revised notes and Wikipedia read","updated":"2026-10-29","human_review":null,"version":{"date":"2026-10-29"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":159,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":98,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":189,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":128,"in_training_data":false}],"short_url":"https://postcutoff.com/s/hadwiger-conjecture-disproved"}