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

Two central topics of study in combinatorics are the so-called evolution of random graphs, introduced by the seminal work of Erd\H{o}s and R'enyi, 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 $\text{ex}(n,H)$. In this paper, we resolve this problem in the case when $H$ is a clique, extending a classical result of Kolaitis, Pr"omel, and Rothschild. In particular, we prove that for every $r \ge 2$,thereisanexplicitconstant$\theta_r$suchthat,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 $\epsilon$. If $m \ge (1+\varepsilon) m_r$,thenalmostall$K_{r+1}$-free$n$-vertexgraphswith$m$edgesare$r$-partite,whereasif$n \ll m \le (1-\varepsilon)m_r$, then almost all of them are not $r$-partite.