Reducing the upper bound for the Borsuk number in ${\mathbb{R}}^4$ to 8

Abstract

The Borsuk number $b(n)$ of $n$-dimensional Euclidean space ${\mathbb{R}}^n$ is the smallest integer such that any set $F\subset {\mathbb{R}}^n$ of unit diameter can be partitioned into $b(n)$ subsets of strictly smaller diameter. For $n=4$, the best known upper bound $b(4)\le 9$ follows from a construction by M. Lassak (1982). In the present paper, we construct partitions of several variants of the truncated Lassak cover into 8 parts of diameter less than 1, thereby showing that $b(4)\le 8$.

14 pages, 5 figures