Testing subgraphs in large graphs

Let H be a fixed graph with h vertices, let G be a graph on n vertices and suppose that at least εn² edges have to be deleted from it to make it H-free. It is known that in this case G contains at least f (ε, H)n^h copies of H. We show that the largest possible function f (ε, H) is polynomial in ε if and only if H is bipartite. This implies that there is a one-sided error property tester for checking H-freeness, whose query complexity is polynomial in 1/ε, if and only if H is bipartite.