The extremal function for partial bipartite tilings

For a fixed bipartite graph H and given number c, 0<c<1, we determine the threshold T_H(c) which guarantees that any n-vertex graph with at edge density at least T_H(c) contains $(1-o(1))c/v(H) n$ vertex-disjoint copies of H. In the proof we use a variant of a technique developed by Komlos~\bcolor{[Combinatorica 20 (2000), 203-218}]