{"id":"openai-math/The-Euclidean-plane-is-not-five-colorable","pdf":"https://github.com/openai/math/blob/main/preprints/The-Euclidean-plane-is-not-five-colorable-September-23-2026/paper.pdf","url":"https://github.com/openai/math/tree/main/preprints/The-Euclidean-plane-is-not-five-colorable-September-23-2026","tags":["discrete-geometry","chromatic-number","combinatorics","plane-coloring"],"score":0,"title":"The Euclidean plane is not five-colorable","domain":"Combinatorics","family":"158","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.CO","summary":"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.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":3,"published":"2026-09-23","reactions":{},"discussion":[{"url":"https://gilkalai.wordpress.com/2026/10/07/updates-sharing-ai-progress-on-mathematics-amazing-and-my-lecture-plans/#:~:text=raises%20the%20lower%20bound%20to%20six%2C%20leaving%20six%20and%20seven%20as%20the%20two%20possible%20answers","date":"2026-10-07","kind":"web","site":"Combinatorics and more","text":"raises the lower bound to six, leaving six and seven as the two possible answers. The Kakeya problem. My post on Hong Wang and Joshua Zahl’s breakthrough discussed the three-dimensional Kakeya set conjecture and the stronger maximal-function problem. The release announces the maximal conjecture in…","added":"2026-10-24","title":"Updates: Sharing AI progress on mathematics (amazing!); and my lecture plans"},{"url":"https://mathoverflow.net/questions/516012/what-is-an-example-of-a-finite-unit-distance-graph-that-isnt-5-colorable","date":"2026-10-13","kind":"web","site":"mathoverflow.net","text":"In one of OpenAI's recent papers , there is a claim that the unit distance graph of the euclidean plane isn't 5-colorable , which is major progress on the Hadwiger–Nelson problem ( assuming the paper isn't retracted ). This result implies, by the De Bruijn–Erdős theorem or the contrapositive of…","added":"2026-10-24","title":"What is an example of a finite unit-distance graph that isn't 5-colorable?","author":"Christopher King"},{"url":"https://www.youtube.com/watch?v=QNSbI3mrrhc","kind":"web","site":"youtube.com","text":"Enjoy the videos and music you love, upload original content, and share it all with friends, family, and the world on YouTube.","added":"2026-10-24","title":"OpenAI Claimed 722 Math Breakthroughs. Did Any Hold Up?","author":"Nontrivial"}],"formalized":true,"family_title":"The Euclidean plane cannot be colored with five colors"}