On the Bolloba's-Eldridge conjecture for bipartite graphs

Let G be a simple graph on n vertices. A conjecture of Bollobas and Eldridge [5] asserts that if 6(G) >= kn-1/k=1 then G contains any n vertex graph H with Delta(H) = k. We prove a strengthened version of this conjecture for bipartite, bounded degree 11, for sufficiently large n. This is the first result on this conjecture for expander graphs of arbitrary (but bounded) degree. An important tool for the proof is a new version of the Blow-Up Lemma.