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

TL;DR

This work investigates the Borsuk problem for subsets of the Boolean cube $\{0,1\}^n$, focusing on whether every subset of bounded diameter can be partitioned into at most $n+1$ parts of smaller diameter. It extends the known result from $n\le 9$ to $n=10$ by employing a universal $(n,k)$-cover framework and analyzing the chromatic numbers of diameter graphs $G_{n,k}$, using heuristics and the kissat SAT solver to manage the computational complexity. The authors present two main case studies for $n=10$: a $k=4$ covering system $\mathcal{C}^*_{10,4}$ and a $k=6$ backtracking scheme yielding a covering $\mathcal{T}_{10,6}$, both achieving $\chi(\cdot;k) \le 11$. They conclude that the $(0,1)$-Borsuk conjecture holds for $n=10$ in this discrete setting and discuss the substantial computational effort required, suggesting weighted formulations or specialized algorithms for extending the verification to larger dimensions.

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 \leq 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 $\textbf{kissat}$ was used to cut off the search branches.

Theorems & Definitions (9)

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• Definition 2
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