The extremal function for partial bipartite tilings

For a fixed bipartite graph H and given α ∈ ( 0 , 1 ) , we determine the threshold T H ( α ) which guarantees that any n -vertex graph with at least T H ( α ) n 2 edges contains ( 1 − o ( 1 ) ) α v ( H ) n vertex-disjoint copies of H . In the proof, we use a variant of a technique developed by Komlós [J. Komlós, Tiling Turán theorems, Combinatorica 20 (2) (2000) 203–218].