Rainbow sets in the intersection of two matroids

Abstract

Given sets $F_1,\dots ,F_n$, 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 $F_1,\dots ,F_n$ 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 $n-\sqrt{n}$.

References in corpus (1)

• Rainbow matchings and partial transversals of Latin squares

Cited by in corpus (1)

• A matroidal generalization of results of Drisko and Chappell