The number of K s,t -free graphs

Abstract

Denote by fn (H) the number of (labeled) H-free graphs on a fixed vertex set of size n. Erdős conjectured that, whenever H contains a cycle, f n ( H ) = 2 ( 1 + o ( 1 ) ) ex ( n , H ) , yet it is still open for every bipartite graph, and even the order of magnitude of log2 fn (H) was known only for C4, C6, and K3,3. We show that, for all s and t satisfying 2 ⩽ s ⩽ t , f n ( K s , t ) = 2 O ( n 2 − 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, Ks,t) is known. Our methods allow us to prove, among other things, that there is a positive constant c such that almost all K2,t-free graphs of order n have at least 1/12 · ex(n, K2,t) and at most (1 − c) ex(n, K2,t) edges. Moreover, our results have some interesting applications to the study of some Ramsey- and Turán-type problems.