{"id":"openai-math/A-nine-dimensional-counterexample-to-Borsuks-covering-assertion","pdf":"https://github.com/openai/math/blob/main/preprints/A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026/paper.pdf","url":"https://github.com/openai/math/tree/main/preprints/A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026","tags":["discrete-geometry","convex-geometry","borsuk-problem"],"score":0,"title":"A nine-dimensional counterexample to Borsuk's covering assertion","domain":"Combinatorics","family":"156","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.CO","summary":"The compact set of rank-one orthogonal projectors on ℝ 4 , with the Frobenius metric, cannot be covered by ten sets of strictly smaller diameter. It therefore gives a counterexample to Borsuk's conjecture in dimension nine.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":1,"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=a%20compact%20set%20in%20nine-dimensional%20Euclidean%20space%20that%20cannot%20be%20covered%20by%20ten%20sets%20of%20smaller%20diameter","date":"2026-10-07","kind":"web","site":"Combinatorics and more","text":"a compact set in nine-dimensional Euclidean space that cannot be covered by ten sets of smaller diameter. Hadwiger’s conjecture. In Combinatorics News I discussed Hadwiger’s conjecture, which relates graph coloring to complete minors, while reporting the disproof of its odd-minor strengthening.…","added":"2026-10-24","title":"Updates: Sharing AI progress on mathematics (amazing!); and my lecture plans"}],"formalized":true,"family_title":"Borsuk's conjecture fails in dimension nine"}