The typical structure of sparse Kr+1-free graphs

Two central topics of study in combinatorics are the so-called evolution of random graphs, introduced by the seminal work of Erdős and Rényi, and the family of H-free graphs, that is, graphs which do not contain a subgraph isomorphic to a given (usually small) graph H. A widely studied problem that lies at the interface of these two areas is that of determining how the structure of a typical H-free graph with n vertices and m edges changes as m grows from 0 to ex(n,H). In this paper, we resolve this problem in the case when H is a clique, extending a classical result of Kolaitis, Prömel, and Rothschild. In particular, we prove that for every r> 2, there is an explicit constant θr such that, letting mr = θrn 2 − 2 r+2 (logn)1/[( r+1 2)−1], the following holds for every positive constant ε. If m> (1 + ε)mr, then almost all Kr+1-free n-vertex graphs with m edges are r-partite, whereas if n m 6 (1 − ε)mr, then almost all of them are not r-partite