Combinatorics·OpenAI Math·Result 156·Lean formalisation listed
Borsuk's conjecture fails in dimension nine
One of the 372 results in OpenAI's public mathematics catalogue, filed under combinatorics: 1 preprint, the latest dated 23 September 2026. This page quotes OpenAI's own summary of the claim, links the paper and abstract, and will embed the animation.
Latest paper
A nine-dimensional counterexample to Borsuk's covering assertion
The animation for this result has not been released yet. Subscribe on YouTube to catch it.
What OpenAI claims
Constructs a compact subset of ℝ9 that cannot be covered by ten sets of strictly smaller diameter, disproving Borsuk's covering assertion already in dimension nine. The example consists of rank-one orthogonal projectors onto lines in ℝ4, with the Frobenius metric. (Lean)
Paper
A nine-dimensional counterexample to Borsuk's covering assertion
The compact set of rank-one orthogonal projectors on ℝ4, 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.
Status
Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.
Related results in Combinatorics
- 157Graph coloring, clique minors, and Colin de Verdière invariantsLean
- 158The Euclidean plane cannot be colored with five colorsVideoLean
- 159Erdős’s reciprocal-sum conjecture and quasipolynomial Szemerédi boundsVideoLean
- 160Superexponential van der Waerden numbersLean
- 161Counterexamples to Sidorenko’s conjecture and the forcing conjectureLean
- 162Counterexamples to Ryser’s covering conjectureLean
