Borsuk's partition problem

DisprovedGeometry

Statement

Can every bounded subset $E \subset \mathbb{R}^n$ of positive diameter $D = \sup_{x,y \in E} \|x - y\| > 0$ be partitioned into at most $n + 1$ subsets $E_1, \dots, E_{n+1}$ such that the diameter of each piece $E_i$ is strictly less than $D$ ?

Disproved in general by Jeff Kahn and Gil Kalai in 1993. Using the Frankl–Wilson intersection theorem on $(0,1)$ -vectors, they constructed finite point sets in $\mathbb{R}^n$ requiring at least $(1.2)^{\sqrt{n}}$ pieces of smaller diameter, yielding an explicit counterexample for $n = 1{,}325$ and for all $n > 2{,}014$ . Subsequent work steadily reduced the smallest dimension where Borsuk's conjecture fails: Andriy V. Bondarenko (2013) used two-distance sets from strongly regular graphs to find a counterexample in $\mathbb{R}^{65}$ , Thomas Jenrich and Andries E. Brouwer (2014) found a 352-point counterexample in $\mathbb{R}^{64}$ , and a 2026 preprint by Yibo Ji verifies an AI-generated 321-point counterexample in $\mathbb{R}^{63}$ .

Kahn–Kalai's disproof of Borsuk's conjecture via the Frankl–Wilson theoremJeff Kahn and Gil Kalai; dimension lowered by Andriy V. Bondarenko, Thomas Jenrich, Andries E. Brouwer, and Yibo Ji, 2013Difficulty 4/5Advanced

References

Karol Borsuk (1933). Drei Sätze über die n-dimensionale euklidische Sphäre · DOI:10.4064/fm-20-1-177-190
Jeff Kahn, Gil Kalai (1993). A counterexample to Borsuk's conjecture · DOI:10.1090/S0273-0979-1993-00398-7 · arXiv:math/9307229v1
Andriy V. Bondarenko (2014). On Borsuk's conjecture for two-distance sets · DOI:10.1007/s00454-014-9579-4 · arXiv:1305.2584v2
Thomas Jenrich, Andries E. Brouwer (2014). A 64-dimensional counterexample to Borsuk's conjecture · DOI:10.37236/4069 · arXiv:1308.0206v3
Yibo Ji (2026). An AI Generated Counterexample to Borsuk Problem in Dimension 63 · arXiv:2608.12561v1 [preprint, not peer-reviewed]

History of the problem

1933Karol Borsuk proves the conjecture for $n = 2$ (any planar set of diameter $D$ can be covered by three sets of diameter at most $\frac{\sqrt{3}}{2} D < D$ ) and poses the general $n$ -dimensional question at the end of his paper on the $n$ -sphere.
1946Hugo Hadwiger proves that every bounded subset of $\mathbb{R}^n$ with a smooth boundary (where every boundary point has a unique supporting hyperplane) can be partitioned into $n + 1$ parts of smaller diameter.
1955H. G. Eggleston gives a complete proof for $n = 3$ (following an earlier 1947 announcement by Julian Perkal), showing that every bounded subset of $\mathbb{R}^3$ can be partitioned into $4$ pieces of smaller diameter.
1993Jeff Kahn and Gil Kalai disprove Borsuk's conjecture in high dimensions, proving that the minimum number of smaller-diameter pieces needed in $\mathbb{R}^n$ grows exponentially as at least $(1.2)^{\sqrt{n}}$ .
2013Andriy V. Bondarenko constructs a two-distance counterexample in $\mathbb{R}^{65}$ from the strongly regular $G_2(4)$ graph, which Thomas Jenrich and Andries E. Brouwer (2014) refine to a 352-point counterexample in $\mathbb{R}^{64}$ requiring at least $71 > 65$ parts.