Problem 505

Is every set of diameter $1$ in $\mathbb{R}^n$ the union of at most $n+1$ sets of diameter $<1$ ?

Geometry

Wiki page

Status

SolvedDisproved

An accepted solution exists. The statement is false.

Disproved. The site marks the problem disproved, crediting Kahn and Kalai, with Brouwer and Jenrich for the smallest dimension it records, and flags a Lean formalization.

Bo33: Borsuk, K., Drei Sätze über die n-dimensionale euklidische Sphäre. Fund. Math. 20 (1933), 177-190.
BrJe14: Jenrich, Thomas and Brouwer, Andries E., A 64-dimensional counterexample to Borsuk's conjecture. Electron. J. Combin. (2014), Paper 4.29, 3.
Eg55: Eggleston, H. G., Covering a three-dimensional set with sets of smaller diameter. J. London Math. Soc. (1955), 11-24.
Er81b: Erdős, P., My Scottish Book 'Problems'. The Scottish Book (1981), 27-35 (page numbers are given for the 2nd edition of The Scottish Book).
KaKa93: Kahn, Jeff and Kalai, Gil, A counterexample to Borsuk's conjecture. Bull. Amer. Math. Soc. (N.S.) (1993), 60-62.
Sc88: Schramm, Oded, Illuminating sets of constant width. Mathematika (1988), 180-189.
OAI26: OpenAI, A nine-dimensional counterexample to Borsuk's covering assertion. OpenAI Math Release preprint (23 September 2026), https://github.com/openai/math, folder preprints/A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026.

Problem 505

Status

Proof claims

Discussion