Forbidden induced bipartite graphs

Given a fixed bipartite graph H, we study the asymptotic speed of growth of the number of bipartite graphs on n vertices which do not contain an induced copy of H. Whenever H contains either a cycle or the bipartite complement of a cycle, the speed of growth is equation image. For every other bipartite graph except the path on seven vertices, we are able to find both upper and lower bounds of the form equation image. In many cases we are able to determine the correct value of c.