A common generalization to strengthenings of Drisko's Theorem for intersections of two matroids

TL;DR

This paper generalizes Drisko's theorem by proving the existence of a large common independent rainbow set in two matroids, extending previous results in bipartite graphs and matroid intersections.

Contribution

It introduces a unified generalization that strengthens existing theorems on rainbow matchings and matroid intersections.

Findings

Existence of a partial rainbow set of size n in two matroids.

Generalization of rainbow matching results for bipartite graphs.

Extension of Kotlar and Ziv's intersection theorem.

Abstract

Let $\mathcal{M}$ and $\mathcal{N}$ be two matroids on the same ground set $V$. Let $A_1,\dots,A_{2n-1}$ be sets which are independent in both $\mathcal{M}$ and $\mathcal{N}$, satisfying $|A_i|\geq \textrm{min}(i,n)$ for all $i$. We show that there exists a partial rainbow set of size $n$, which is independent in both $\mathcal{M}$ and $\mathcal{N}$. This is a common generalization of rainbow matching results for bipartite graphs by Aharoni, Berger, Kotlar, and Ziv, and for the intersection of two matroid by Kotlar and Ziv.