Math Conjectures

Four-Dimensional Borsuk Problem

OPENMajorCanonical finite caseProposed 1933 · Canonical special case

Canonical statement

Let b(4)b(4) be the least mm such that every bounded set S⊂R4S\subset\mathbb R^4 of positive diameter can be partitioned into mm subsets, each having diameter strictly smaller than diam⁡(S)\operatorname{diam}(S). Is
b(4)=5? b(4)=5?
View source LaTeX
Let \(b(4)\) be the least \(m\) such that every bounded set
\(S\subset\mathbb R^4\) of positive diameter can be partitioned into
\(m\) subsets, each having diameter strictly smaller than
\(\operatorname{diam}(S)\). Is
\[
  b(4)=5?
\]
DG-014:[Borsuk1933][KahnKalai1993Borsuk][TolmachevVoronov2026Borsuk]
Discrete geometryGeometry and topology

Notes

Borsuk asked in 1933 whether every bounded set of positive diameter in Rd\mathbb R^d can be partitioned into d+1d+1 pieces, each of strictly smaller diameter [Borsuk1933]. Writing b(d)b(d) for the least number of pieces that always suffices, the question here is the first unresolved instance of that problem: is b(4)=5b(4)=5?

The answer to Borsuk's general question is affirmative in dimensions up to three, but Kahn and Kalai showed in 1993 that it fails in high dimensions [KahnKalai1993Borsuk], so b(d)=d+1b(d)=d+1 cannot be taken for granted anywhere it is not proved. In dimension four the regular simplex forces b(4)≥5b(4)\ge 5. For the upper bound, a May 2026 preprint of Tolmachev and Voronov reduced the previously known bound of 99 to 88 [TolmachevVoronov2026Borsuk], so at present 5≤b(4)≤85\le b(4)\le 8.

The case remains open: one must either exhibit a partition scheme showing that every bounded four-dimensional set splits into five pieces of smaller diameter, or find a four-dimensional set requiring six or more.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.