Simonovits's theorem in random graphs

TL;DR

This paper extends Simonovits's theorem to random graphs, showing that the largest $H$-free subgraph in $G_{n,p}$ is asymptotically the largest $r$-partite subgraph under certain conditions, partially confirming a conjecture.

Contribution

It generalizes Simonovits's theorem to the binomial random graph $G_{n,p}$ for a class of graphs $H$, providing conditions on $p$ and resolving a conjecture.

Findings

Largest $H$-free subgraph in $G_{n,p}$ is $r$-partite for certain $p$

Conditions on $p$ are nearly optimal and depend on $H$

Partially confirms a conjecture of DeMarco and Kahn

Abstract

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.