Perfect matchings in uniform hypergraphs with large minimum degree

A perfect matching in a k -uniform hypergraph on n vertices, n divisible by k , is a set of n / k disjoint edges. In this paper we give a sufficient condition for the existence of a perfect matching in terms of a variant of the minimum degree. We prove that for every k ≥ 3 and sufficiently large n , a perfect matching exists in every n -vertex k -uniform hypergraph in which each set of k − 1 vertices is contained in n / 2 + Ω ( log n ) edges. Owing to a construction in [D. Kühn, D. Osthus, Matchings in hypergraphs of large minimum degree, J. Graph Theory 51 (1) (2006) 269–280], this is nearly optimal. For almost perfect and fractional perfect matchings we show that analogous thresholds are close to n / k rather than n / 2 .