A better bound on the size of rainbow matchings

Aharoni and Howard conjectured that, for positive integers $n,k,t$ with $n\ge k$ and $n\ge t$, if $F_1,\ldots, F_t\subseteq {[n]\choose k}$ such that $|F_i|>{n\choose k}-{n-t+1\choose k}$ for $i\in [t]$ then there exist $e_i\in F_i$ for $i\in [t]$ such that $e_1,\ldots,e_t$ are pairwise disjoint. Huang, Loh, and Sudakov proved this conjecture for $t<n/(3k^2)$. In this paper, we show that this conjecture holds for $t\le n/(2k)$ and $n$ sufficiently large.