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. Kuhn, 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.