Matroids, intersecting bases, and Borsuk property

TL;DR

This paper explores the Borsuk property within the context of matroids, providing combinatorial criteria and examples, and linking geometric and topological concepts through matroid polytopes and Kneser graphs.

Contribution

It introduces a combinatorial approach to the Borsuk property for matroid polytopes and identifies sufficient conditions involving disjoint bases, along with counterexamples.

Findings

Matroids with two disjoint bases satisfy the Borsuk property.

Counterexamples show the condition is not necessary.

Kneser graphs are key tools in proofs.

Abstract

A subset $S$ of $\mathbb R^d$ has the Borsuk property if it can be decomposed into at most $d+1$ parts of diameter smaller than $S$. This is an important geometric property, inspired by a conjecture of Borsuk from the 1930s, which has attracted considerable attention over the years. In this paper, we define and investigate the Borsuk property for matroids, providing a purely combinatorial approach to the Borsuk property for matroid polytopes, a well-studied family of $(0,1)$-polytopes associated with matroids. We show that a sufficient condition for a matroid -- and thus its matroid polytope -- to have the Borsuk property is that the matroid or its dual has two disjoint bases. However, we show that this condition is not necessary by exhibiting infinite families of matroids having the Borsuk property and yet being such that every two bases intersect and every two cobases intersect. Kneser graphs, which form an important object from topological combinatorics, play a crucial role in most proofs.