The Borsuk Problem for Subsets of the Vertices of the 10-Dimensional Boolean Cube

Abstract

In the papers Ziegler(2001) and Goldstein(2012) it was previously shown that any subset of the Boolean cube $S\subset \{0,1{\}}^n$ for $n\le 9$ can be partitioned into $n+1$ parts of smaller diameter, i.e., the Borsuk conjecture holds for such subsets. In this paper, it is shown that this is also true for $n=10$; however, the complexity of the computational verification increases significantly. In order to perform the computations in a reasonable time, several heuristics were developed to reduce the search tree. The SAT solver $\text{kissat}$ was used to cut off the search branches.