Rainbow sets in the intersection of two matroids

TL;DR

This paper proves that for two matroids with certain disjoint sets, there exists a large rainbow set of size at least n minus the square root of n, advancing the understanding of rainbow sets in matroid intersections.

Contribution

It establishes a lower bound of n - √n for the size of rainbow sets in the intersection of two matroids, improving upon the previously conjectured size of n-1.

Findings

Existence of rainbow sets of size at least n - √n in matroid intersections.

Progress towards the conjecture of rainbow set size n-1.

Application of ideas from Woolbright and Brower-de Vries-Wieringa.

Abstract

Given sets $F_1, \ldots ,F_n$, a {\em partial rainbow function} is a partial choice function of the sets $F_i$. A {\em partial rainbow set} is the range of a partial rainbow function. Aharoni and Berger \cite{AhBer} conjectured that if $M$ and $N$ are matroids on the same ground set, and $F_1, \ldots ,F_n$ are pairwise disjoint sets of size $n$ belonging to $M \cap N$, then there exists a rainbow set of size $n-1$ belonging to $M \cap N$. Following an idea of Woolbright and Brower-de Vries-Wieringa, we prove that there exists such a rainbow set of size at least $n-\sqrt{n}$.