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

TL;DR

This paper characterizes the typical structure of $K_{r+1}$-free graphs with a given number of edges, showing a phase transition around a critical edge count where graphs switch from being mostly non-$r$-partite to mostly $r$-partite.

Contribution

It extends classical results by precisely identifying the threshold for the emergence of $r$-partiteness in $K_{r+1}$-free graphs as the number of edges varies.

Findings

Identifies an explicit threshold $m_r$ for the transition in graph structure.

Shows that above $m_r$, almost all such graphs are $r$-partite.

Below $m_r$, almost all such graphs are not $r$-partite.

Abstract

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 $r \ge 2$, 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 $m \ge (1+\varepsilon) m_r$, then almost all $K_{r+1}$-free $n$-vertex graphs with $m$ edges are $r$-partite, whereas if $n \ll m \le (1-\varepsilon)m_r$, then almost all of them are not $r$-partite.