Matroid colorings of KKM covers

TL;DR

This paper extends the KKM theorem to matroid-colored set coverings of polytopes, generalizing previous colorful KKM theorems and their sparse variants, thus broadening the theoretical framework of combinatorial topology.

Contribution

It introduces a matroid coloring version of the KKM theorem, unifying and generalizing several existing colorful and sparse KKM results.

Findings

Proves a new matroid-colored KKM theorem for polytopes.

Generalizes Gale's colorful KKM theorem.

Extends recent sparse-colorful KKM variants.

Abstract

We prove a KKM-type theorem for matroid colored families of set coverings of a polytope. This generalizes Gale's colorful KKM theorem as well as recent sparse-colorful variants by Sober\'on, and McGinnis and Zerbib.