Oriented matroids and Ky Fan's theorem

TL;DR

This paper generalizes Ky Fan's theorem using oriented matroids, extending Lovasz's matroid-based generalization of Sperner's lemma, and demonstrates how classical results can be derived as special cases.

Contribution

It introduces an oriented matroid framework for Ky Fan's theorem, broadening the scope of combinatorial topology and matroid theory applications.

Findings

Ky Fan's theorem admits an oriented matroid generalization.

Classical Ky Fan's theorem is recovered as a special case.

The approach links matroid theory with topological combinatorics.

Abstract

L. Lovasz has shown that Sperner's combinatorial lemma admits a generalization involving a matroid defined on the set of vertices of the associated triangulation. Inspired by this result we prove that classical Ky Fan's theorem admits an oriented matroid generalization of similar nature. Ky Fan's theorem is obtained as a corollary if the underlying oriented matroid is chosen to be the alternating matroid C^{m,r} .