Completing partial packings of bipartite graphs

Given a bipartite graph H and an integer n, let f(n;H) be the smallest integer such that any set of edge disjoint copies of H on n vertices can be extended to an H-design on at most n+f(n;H) vertices. We establish tight bounds for the growth of f(n;H) as n→∞. In particular, we prove the conjecture of Füredi and Lehel (2010) [4] that f(n;H)=o(n). This settles a long-standing open problem.