A nine-dimensional counterexample to Borsuk's covering assertion
Source: github.com/openai/math/tree/main/preprints/A-nine-dimension…
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.
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One of the 700 or so papers released by Open AI on October 6, 2026 asserts that f(9)>10 . This is a major progress on the Borsuk’s problem. (Of course, the proof needs to be carefully checked. The main theorem has also been formalized in Lean.) The construction itself is one of the constructions…
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Updates: Sharing AI progress on mathematics (amazing!); and my lecture plansCombinatorics and moreGil KalaiOct 7, 2026
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.…