A better bound on the size of rainbow matchings

Hongliang Lu, Yan Wang, Xingxing Yu

Aharoni and Howard conjectured that, for positive integers $n,k,t$ with $n\ge k$ and $n\ge t$, if $F_1,\dots ,F_t\subseteq \left(\genfrac{}{}{0ex}{}{[n]}{k}\right)$ such that $|F_i|>{n\choose k}-{n-t+1\cho…