Near Perfect Matchings ink-Uniform Hypergraphs

Abstract

LetHbe ak-uniform hypergraph onnvertices wherenis a sufficiently large integer not divisible byk. We prove that if the minimum (k− 1)-degree ofHis at least ⌊n/k⌋, thenHcontains a matching with ⌊n/k⌋ edges. This confirms a conjecture of Rödl, Ruciński and Szemerédi [13], who proved that minimum (k− 1)-degreen/k+O(logn) suffices. More generally, we show thatHcontains a matching of sizedif its minimum codegree isd<n/k, which is also best possible.