A combinatorial version of the colorful Caratheodory theorem

TL;DR

This paper extends Barany's colorful Caratheodory theorem to oriented and matroids, establishing conditions under which certain circuits exist within independent sets, broadening the theorem's applicability.

Contribution

It introduces a combinatorial extension of the colorful Caratheodory theorem involving oriented and matroids with specific rank conditions.

Findings

Established a new condition linking positive circuits and matroid independence.

Extended the theorem's applicability to broader combinatorial structures.

Provided a proof under the specified rank and circuit conditions.

Abstract

We give the following extension of Barany's colorful Caratheodory theorem: Let M be an oriented matroid and N a matroid with rank function r, both defined on the same ground set V and satisfying rank(M) < rank(N). If every subset A of V with r(V - A) < rank (M) contains a positive circuit of M, then some independent set of N contains a positive circuit of M.