Result 156, Combinatorics

Borsuk's conjecture fails in dimension nine

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 formalization Disproof or counterexample

The bigger picture

Why it matters

Borsuk's covering assertion proposes that any bounded shape in n dimensions can be covered by n plus one pieces of smaller diameter. The manuscript claims that, even in nine dimensions, ten pieces need not suffice.

What changes?

The manuscript reports a compact set in nine-dimensional Euclidean space that cannot be covered by ten sets of strictly smaller diameter. Diameter means the greatest distance between points of this compact set. Its points are rank-one orthogonal projectors: matrices that project vectors onto lines through the origin in four-dimensional space. Their distances use the Frobenius metric, calculated as the square root of the sum of squared differences between corresponding matrix entries.

What does that help mathematicians do?

The claimed obstruction applies to every possible ten-set cover, not just to a particular way of cutting up the example. Consequently, any cover of this set by pieces of strictly smaller diameter would require at least eleven pieces. This does not establish that eleven suffice. It rules out a universal dimension-plus-one covering bound already in dimension nine, using a family of geometric projection matrices.

Are there practical applications?

The immediate value is foundational: the example connects lines in four-dimensional space with a covering obstruction in nine-dimensional Euclidean geometry. Researchers studying diameter reduction can use this claimed counterexample to test proposed covering principles and identify where additional assumptions are needed. The result is an obstruction, not an algorithm for finding covers or a demonstrated practical application.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

A nine-dimensional counterexample to Borsuk's covering assertion

September 23, 2026 18 pages Main result formalized in Lean

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.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/156.md.

Borsuk's conjecture fails in dimension nine

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 Rd\mathbb R^d can be covered by d+1d+1 sets of strictly smaller diameter. The formalized counterexample is the compact set of rank-one orthogonal projectors onto lines in R4\mathbb R^4, with the Frobenius metric. It lies in the nine-dimensional affine space of trace-one symmetric matrices, has diameter 2\sqrt2, and cannot be covered by ten arbitrary sets of smaller diameter.

Comparator links

Result Comparator statement
Nine-dimensional Borsuk counterexample BorsukNine.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 7 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.