Nearly perfect matchings in regular simple hypergraphs

For every fixed k ≥ 3 there exists a constant c_k with the following property. Let H be a k-uniform, D-regular hypergraph on N vertices, in which no two edges contain more than one common vertex. If k > 3 then H contains a matching covering all vertices but at most c_kND^-1/(k-1). If k = 3, then H contains a matching covering all vertices but at most c₃ND^-1/2ln^3/2D. This improves previous estimates and implies, for example, that any Steiner Triple System on N vertices contains a matching covering all vertices but at most 0(N^1/2ln^3/2N), improving results by various authors.