The number of K_s,t-free graphs

Denote by f_n (H) the number of (labeled) H-free graphs on a fixed vertex set of size n. Erdos conjectured that, whenever H contains a cycle, f_n(H) = 2^{(1+0(1))ex(n,h)}, yet it is still open for every bipartite graph, and even the order of magnitude of log₂ f _n (H) was known only for C₄, C₆, and K _3,3. We show that, for all s and t satisfying 2≤s≤t, f _n(K_s,t=2^O(n2-1/s), which is asymptotically sharp for those values of s and t for which the order of magnitude of the Turán number ex(n, K_s,t) is known. Our methods allow us to prove, among other things, that there is a positive constant c such that almost all K_2,t-free graphs of order n have at least 1/12 · ex(n, K_2,t) and at most (1 - c) ex(n, K_2,t) edges. Moreover, our results have some interesting applications to the study of some Ramsey- and Turán-type problems.