Let $H$ be a graph with $\chi(H) = r+1$. Simonovits's theorem states that, if $H$ is edge-critical, the unique largest $H$-free subgraph of $K_n$ is its largest $r$-partite subgraph, provided that $n$ is sufficiently large. We show that the same holds with $K_n$ replaced by the binomial random graph $G_{ n,p}$ whenever $H$ is also strictly $2$-balanced and $p \ge (\theta_H+o(1)) n^{ -\frac{ 1}{ m_2(H)}} (\log n)^{ \frac{ 1}{ e_H-1}}$ for some explicit constant $\theta_H$, which we believe to be optimal. This (partially) resolves a conjecture of DeMarco and Kahn.
Categories: math.CO math.PR