Near Perfect Matchings in k-Uniform Hypergraphs

Let H be a k-uniform hypergraph on n vertices where n is a sufficiently large integer not divisible by k. We prove that if the minimum (k - 1)-degree of H is at least [n/k], then H contains a matching with [n/k] edges. This confirms a conjecture of Rödl, Ruciński and Szemerédi [13], who proved that minimum (k - 1)-degree n/k + O(logn) suffices. More generally, we show that H contains a matching of size d if its minimum codegree is d < n/k, which is also best possible.