Nearly perfect matchings in regular simple hypergraphs

For every fixed k ≥ 3 there exists a constant ck with the following property. Let H be a kuniform, 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 ckND. If k = 3, then H contains a matching covering all vertices but at most c3ND lnD. 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 O(N lnN), improving results by various authors.