Sharp bounds for rainbow matchings in hypergraphs

Suppose that we are given matchings $M_1,\text{…}.,M_N$ of size $t$ in some $r$-uniform hypergraph, and let us think of each matching having a different color. How large does $N$ need to be (in terms of $t$ and $r$) such that we can always find a rainbow matching of size $t$? This problem was first introduced by Aharoni and Berger, and has since been studied by several different authors. For example, Alon discovered an intriguing connection with the Erdős–Ginzburg–Ziv problem from additive combinatorics, which implies certain lower bounds for $N$. For any fixed uniformity $r⩾3$, we answer this problem up to constant factors depending on $r$, showing that the answer is on the order of $t^r$. Furthermore, for any fixed $t$ and large $r$, we determine the answer up to lower order factors. We also prove analogous results in the setting where the underlying hypergraph is assumed to be $r$-partite. Our results settle questions of Alon and of Glebov–Sudakov–Szabó.