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

TL;DR

Problem: generalizes Drisko's rainbow matching to the intersection of two matroids by using $2n-1$ independent sets $A_i$ with $|A_i|\ge \min(i,n)$. Approach: a topological-combinatorial framework based on simplicial complexes, the $\eta$-invariant, and a matchability theorem to enforce a common independent diagonal. Key contribution: proves the main theorem that there exists a partial rainbow set of size $n$ independent in both matroids, unifying the matroid-intersection and bipartite rainbow matching results. Significance: provides a common generalization and a toolkit (Meshulam-type bounds, Aharoni-Berger criterion) for related rainbow and transversal problems.

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.

Theorems & Definitions (12)

• Conjecture 1.1: Berger-Aharoni aharoni2009rainbow
• Theorem 1.2: Aharoni-Berger aharoni2009rainbow
• Theorem 1.3: Drisko drisko1998transversals
• Theorem 1.4: Chappel chappell1999matroid
• Theorem 1.5: Kotlar-Ziv kotlar2015rainbow
• Theorem 1.6: Aharoni et. al. aharoni2018degree
• Theorem 1.7: Main theorem
• Theorem 2.1
• Theorem 2.2: Aharoni-Berger aharoni2006intersection
• Lemma 3.1