The typical structure of sparse $K_{r+1}$-free graphs

Abstract:

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 $\mathrm {ex}(n,H)$. 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 $r \geqslant 2$, there is an explicit constant $\theta _r$ such that, letting $m_r = \theta _r n^{2-\frac {2}{r+2}} (\log n)^{1/\left [\binom {r+1}{2}-1\right ]}$, the following holds for every positive constant $\varepsilon$. If $m \geqslant (1+\varepsilon ) m_r$, then almost all $K_{r+1}$-free $n$-vertex graphs with $m$ edges are $r$-partite, whereas if $n \ll m \leqslant (1- \varepsilon )m_r$, then almost all of them are not $r$-partite.