Rainbow matchings in $k$-partite hypergraphs

Abstract

In this paper, we prove a conjecture of Aharoni and Howard on the existence of rainbow (transversal) matchings in sufficiently large families ${\mathcal{F}}_1,\dots ,{\mathcal{F}}_s$ of tuples in $\{1,\dots ,n{\}}^k$, provided $s\ge 470.$