On a Hypergraph Matching Problem

Let H = (V,E) be an r-uniform hypergraph and let F ⊂ 2 . A matching M of H is (α,F)perfect if for each F ∈ F , at least α|F | vertices of F are covered by M . Our main result is a theorem giving sufficient conditions for an r-uniform hypergraph to have a (1− ,F)-perfect matching. As a special case of our theorem we obtain the following result. Let K(n, r) denote the complete r-uniform hypergraph with n vertices. Let t and r be fixed positive integers where t ≥ r ≥ 2. Then, K(n, r) can be packed with edge-disjoint copies of K(t, r) such that each vertex is incident with only o(nr−1) unpacked edges. This extends a result of Rödl [9].