Borsuk’s conjecture in dimension four

Can every bounded set of points in four-dimensional space be split into five groups, each with a strictly smaller span? The span, called diameter, is the least upper bound on distances between points. Start with the three corners of an equilateral triangle: putting any two corners together keeps the original span, so three smaller-span groups are needed. In four dimensions, five equally separated points similarly require five groups. The question is whether five always suffice, even for infinite sets.[1] [2]

Can every bounded set of positive diameter in four-dimensional Euclidean space be divided into five subsets of strictly smaller diameter? A regular simplex supplies five points whose pairwise distances are all one. A subset containing two of them has diameter one, so these five points need five separate subsets. This explains the lower bound; it does not construct five pieces for an arbitrary set. Write $b_4$ for the least number of pieces sufficient for every eligible set. Diameter uses a supremum, so an infinite set need not contain a pair attaining it. The strict requirement applies to each subset’s supremum, not merely to each of its individual distances.[1] [2]

Five parts can be necessary; it is unknown whether they always suffice.[2]

The lower bound comes from a regular four-simplex. A 2026 preprint reports an eight-piece upper bound, leaving the five-piece target unresolved.[2]

This connects shape with a familiar coloring task: points separated by the full diameter cannot go in the same group. The triangle shows how a few mutual distance constraints can force several groups. The four-dimensional problem asks how those constraints fit together across an entire set, where controlling individual pairs may still leave a group with too large a span.[1]

For a finite set, joining pairs at the full diameter gives a graph: a smaller-diameter piece can contain no joined pair. Conversely, a finite group containing no such pair has smaller diameter, because its largest distance is attained. This is a connection between geometric partitioning and graph coloring. For infinite sets that converse needs care: distances can approach the original diameter without ever attaining it. The universal question therefore links finite distance constraints with the behavior of whole bounded sets.[1]