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Combinatorics · Result 156

Borsuk's conjecture fails in dimension nine

1 paperLean

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.

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Papers 1

First page of A nine-dimensional counterexample to Borsuk's covering assertion

A nine-dimensional counterexample to Borsuk's covering assertion

OpenAI · September 23, 2026

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.

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Cite
@misc{OAI:A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026,
  author = {{OpenAI}},
  title = {{A nine-dimensional counterexample to Borsuk's covering assertion}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026/paper.pdf}{OAI:A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026}},
  year = {2026}
}

What was formalized in Lean

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

Borsuk's conjecture predicts that every bounded subset of \(\mathbb R^d\) can be covered by \(d+1\) sets of strictly smaller diameter. The formalized counterexample is the compact set of rank-one orthogonal projectors onto lines in \(\mathbb R^4\), with the Frobenius metric. It lies in the nine-dimensional affine space of trace-one symmetric matrices, has diameter \(\sqrt2\), and cannot be covered by ten arbitrary sets of smaller diameter.

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ResultComparator statement
Nine-dimensional Borsuk counterexampleBorsukNine.lean

Scope note on GitHub →