{"schema":"postcutoff/event@1","as_of":"2026-10-30T13:42:00+01:00","url":"https://postcutoff.com/e/2026-10-06-openai-plane-not-five-colourable-hadwiger-nelson-claim/","md":"https://postcutoff.com/e/2026-10-06-openai-plane-not-five-colourable-hadwiger-nelson-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-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":[{"n":1,"title":"OpenAI (PDF): The Euclidean plane is not five-colorable (23 Sept 2026)","url":"https://github.com/openai/math/blob/main/preprints/The-Euclidean-plane-is-not-five-colorable-September-23-2026/paper.pdf","type":"paper","group":"primary","domain":"github.com"},{"n":2,"title":"OpenAI Lean scope note for family 158","url":"https://github.com/openai/math/blob/main/lean/docs/158.md","type":"code","group":"primary","domain":"github.com"},{"n":3,"title":"oai-math-revised comparator results and LEAN_CHECKS (EuclideanFiveColor, PlaneColoring accepted; definition caveat)","url":"https://github.com/mathcommunity/oai-math-revised/blob/main/docs/LEAN_CHECKS.md","type":"code","group":"primary","domain":"github.com"},{"n":4,"title":"arXiv 2610.18907: Koshelev, Kupavskii & Voronov, Ordinary and measurable colorings of Euclidean three-space","url":"https://arxiv.org/abs/2610.18907","type":"paper","group":"primary","domain":"arxiv.org"},{"n":5,"title":"GitHub: outsidetext/euclidean-chromatic-transfer (Henry Bass, extensions to R^n)","url":"https://github.com/outsidetext/euclidean-chromatic-transfer","type":"code","group":"primary","domain":"github.com"},{"n":6,"title":"Carlos Toledo: OpenAI's math release, what it claims beyond what it checks (findings G7, G10)","url":"https://carlostoledo.co/reports/openai-math.html","type":"discussion","group":"reactions","domain":"carlostoledo.co"},{"n":7,"title":"Dmitry Rybin on X: 'totally unbelievable alien math?' (7 Oct)","url":"https://x.com/DmitryRybin1/status/2107852898617229724","type":"discussion","group":"reactions","domain":"x.com"},{"n":8,"title":"Joshua Zelinsky on Bluesky (6 Oct)","url":"https://bsky.app/profile/joshuazelinsky.bsky.social/post/3mxak2f4ywk24","type":"discussion","group":"reactions","domain":"bsky.app"},{"n":9,"title":"Thomas Bloom: Erdős problems in the OpenAI proofs (problem #508)","url":"https://www.erdosproblems.com/forum/thread/blog:10","type":"discussion","group":"reactions","domain":"erdosproblems.com"},{"n":10,"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"}],"official":5,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"],"title":"OpenAI's math release claims the plane cannot be 5-coloured without two points at distance one sharing a colour, so its chromatic number is 6 or 7 (Hadwiger–Nelson problem); Lean-formalized, but no finite unit-distance graph is exhibited","summary":"\"The Euclidean plane is not five-colorable\" (OpenAI, 23 Sept 2026, 62 pages; family 158 of the 6 Oct release): \"We prove that every coloring of the Euclidean plane with five colors has a monochromatic unit-distance pair, with no regularity assumption on the color classes. Consequently, the chromatic number of the plane is either six or seven.\" The argument transfers any proper colouring with finitely many colours to a \"weak measurable\" one (readers point to its use of amenable groups, Pontryagin duality and ergodic theory), then gives a geometric obstruction to five colours. Lean challenges EuclideanFiveColor (no proper 5-colouring) and PlaneColoring (a proper 7-colouring) were accepted in the oai-math-revised comparator run; an audit that compared definition bodies also certified the five-colour statement. Koshelev, Kupavskii and Voronov already used the transfer theorem in three dimensions (arXiv 2610.18907).","key_facts":["Manuscript (abstract, verbatim): 'We prove that every coloring of the Euclidean plane with five colors has a monochromatic unit-distance pair, with no regularity assumption on the color classes. Consequently, the chromatic number of the plane is either six or seven.' 62 pages, dated 23 Sept 2026. Theorem 1.1 is proved in ZFC; 'The remaining alternatives six and seven are unresolved'","Structure (paper): 'a transfer from unrestricted proper colorings to a measurable condition, and a geometric obstruction to five labels under that condition. The transfer holds for every finite number of colors.' Because of the de Bruijn–Erdős compactness theorem, 'an unrestricted lower bound can also be established without first displaying a finite graph witnessing it'; the final step reuses the Moser spindle","Before: Nelson (1950) and Isbell gave the bounds 4 and 7; Aubrey de Grey's 2018 finite unit-distance graph (1,581 vertices after correction) raised the lower bound to 5, later shrunk by Heule (553 vertices) and Parts (509); for measurable colour classes Falconer had proved at least 5 colours (as recounted in the paper)","Lean (OpenAI scope note lean/docs/158.md): 'The formalized results prove that five colors do not suffice and that seven colors do suffice. The lower bound applies to arbitrary colorings, with no measurability or continuity assumption.' Both challenges 'accepted' in the oai-math-revised comparator run (25–26 Oct). Caveat recorded there: the EuclideanFiveColor configuration let the comparator check the definition ProperColoring by type only; OpenAI emptied that list on 26 Oct. Carlos Toledo's audit had flagged that this one definition 'carries the whole conclusion', ran a strict body-by-body comparison, and later lists EuclideanFiveColor as certified after the change","No explicit witness: Toledo's audit notes that 'the finite non-5-colorable unit-distance graph is obtained by compactness and never displayed', while certifying the paper's finite Moser-spindle certificate in exact arithmetic","Reactions: Joshua Zelinsky (Bluesky, 6 Oct): 'it doesn't look like the method is a direction that the prior lit used to my knowledge. This is far beyond merely building on existing methods'. Dmitry Rybin (X, 7 Oct, about 261k views): 'I was trying to read the OpenAI proof that chromatic number of a plane is >= 6. But it is totally unbelievable alien math?', adding that the model found that any K-colouring is equivalent to a 'weakly measurable' one. Gabriel Gaster (X, 9 Oct) pointed to its mix of 'Pontryagin duality, ergodic theory, and amenable groups'. Gil Kalai (7 Oct): 'The new manuscript raises the lower bound to six'. Thomas Bloom lists it under Erdős problem #508, noting 'This proof has been formalised'","Follow-ups: Koshelev, Kupavskii and Voronov (arXiv 2610.18907, 11 Oct): 'Using the planar character-law rigidity theorem of OpenAI, we prove that every finite proper coloring of R³ can be replaced by a proper Lebesgue-measurable coloring with the same number of colors', so the ordinary and measurable chromatic numbers of R³ are equal; they used ChatGPT 6 Astra to extract the main ideas of OpenAI's preprint. Henry Bass's GitHub repository extends the transfer theorem to R^n with AI-assisted preprints and Lean"],"key_numbers":[],"tags":["math","combinatorics","discrete-geometry","hadwiger-nelson","chromatic-number-of-the-plane","ergodic-theory","openai-math-release","lean","formal-verification"],"science":{"field":"mathematics","subfield":"combinatorics / discrete geometry (Euclidean Ramsey and colouring problems)","problem":"Hadwiger–Nelson problem: the least number of colours needed so that no two points of the plane at distance one share a colour (bounds 5 ≤ χ ≤ 7 before the release)","result":"Claimed proof that 5 colours never suffice, for arbitrary colour classes, so χ(R²) ∈ {6, 7}; Lean formalization of both bounds","open_since":"1950","ai_system":["OpenAI internal model (math release)"],"human_role":"Released as model output under OpenAI's fixed procedure, with no human author named; no human exposition yet; already used as a tool by human authors (with AI help)","verification":{"key":"peer-review","label":"Peer-reviewed","text":"Lean formalization accepted by an independent comparator run and certified by a stricter definition-body check after OpenAI's 26 Oct configuration fix; unrefereed manuscript; finite witness graph not exhibited"},"status":"pending","shock":"The first improvement of the bound on the plane's chromatic number since de Grey's 2018 graph comes not from a bigger finite graph but from a measure-theoretic transfer theorem that readers called 'alien'."},"body_md":"## What happened\n\nThe Hadwiger–Nelson problem asks how many colours are needed to colour the plane so that points at distance exactly one always get\ndifferent colours. Since 1950 the answer was known to lie between 4 and 7. In 2018 the amateur mathematician Aubrey de Grey raised the\nlower bound to 5 with a finite graph of unit distances that cannot be 4-coloured.\n\nOpenAI's 6 Oct release claims 6, so that only 6 and 7 remain. Instead of a larger finite graph, the 62-page manuscript proves that any\nproper colouring with finitely many colours can be replaced by one with a weak measurability property. It then shows that five colours\nare impossible under that property. Compactness implies that a finite non-5-colourable unit-distance graph exists, but the paper does not\nconstruct one. Both bounds are formalized in Lean. A comparator caveat about how one definition was checked was closed by an\nindependent audit and by OpenAI's 26 Oct configuration fix.\n\n## Why it matters\n\nThe chromatic number of the plane is one of the best-known open problems in combinatorics, and this would be the first step since\n2018. Readers note that the method is new rather than a heavier version of de Grey's approach, and other authors have already reused\nit in three dimensions. The claim is formally checked, so the open questions are whether the formal definitions match the classical\nproblem and what the finite witness looks like.","disputed":[{"label":"Verification","text":"Lean formalization accepted by an independent comparator run and certified by a stricter definition-body check after OpenAI's 26 Oct configuration fix; unrefereed manuscript; finite witness graph not exhibited","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":"2026-10-14-ai-assisted-results-mid-october-2026","url":"https://postcutoff.com/e/2026-10-14-ai-assisted-results-mid-october-2026/","date":"2026-10-14","date_precision":"day","short_title":"Mid-October 2026: arXiv results with disclosed AI help","deck":null,"takeaway":"The 13 Oct arXiv listing alone brought a human write-up of GPT-6 Astra's rational exponents proof, Ulam's Lie group question answered with an Astra construction, a better-than-2 k-median algorithm, and about a dozen smaller results whose AI statements range from 'suggested by ChatGPT' to 'GPT 6 Pro was used … to obtain all other new results'.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":20,"official":19,"filed":"2026-10-14","updated":"2026-10-15","orgs":["OpenAI","Anthropic","various mathematicians"]},{"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-hadwiger-conjecture-counterexample-claim","url":"https://postcutoff.com/e/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":13,"official":10,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]}],"people":[{"id":"thomas-bloom","name":"Thomas Bloom","url":"https://postcutoff.com/person/thomas-bloom/"}],"posts":[{"id":"x-dmitryrybin1-2107852898617229724","title":"Rybin on OpenAI's proof that the plane is not 5-colourable: 'totally unbelievable alien math?'","url":"https://postcutoff.com/p/x-dmitryrybin1-2107852898617229724/"}],"videos":[],"models":[],"changes":[{"date":"2026-10-29","type":"filed","text":"Created (CONTENTS entry, PDF (62 pp), Lean scope note, comparator results and LEAN_CHECKS, Bloom, Kalai, posts on X and Bluesky fetched via fxtwitter and the public Bluesky API, Koshelev–Kupavskii–Voronov abstract)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-29","run":null,"sources_read":null,"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/chromatic-number-plane-6"}