Near perfect matchings in k-uniform hypergraphs II

Suppose k lØ n and H is an n-vertex k-uniform hypergraph. A near perfect matching in H is a matching of size [n/k]. We give a divisibility barrier construction that prevents the existence of near perfect matchings in H. This generalizes the divisibility barrier for perfect matchings. We give a conjecture on the minimum d-degree threshold forcing a (near) perfect matching in H which generalizes a well-known conjecture on perfect matchings. We also verify our conjecture for various cases. Our proof makes use of the lattice-based absorbing method that we used recently to solve two other problems on matching and tilings for hypergraphs.