The extremal function for partial bipartite tilings

For a fixed bipartite graph H and given @a@?(0,1), we determine the threshold T"H(@a) which guarantees that any n-vertex graph with at least T"H(@a)n2 edges contains (1-o(1))@av(H)n vertex-disjoint copies of H. In the proof, we use a variant of a technique developed by Komlos [J. Komlos, Tiling Turan theorems, Combinatorica 20 (2) (2000) 203-218].