Chromatic thresholds in sparse random graphs

The chromatic threshold $δ_χ(H,p)$ of a graph $H$ with respect to the random graph $G(n,p)$ is the infimum over $d > 0$ such that the following holds with high probability: the family of $H$-free graphs $G \subset G(n,p)$ with minimum degree $δ(G) \ge dpn$ has bounded chromatic number. The study of $δ_χ(H) :=δ_χ(H,1)$ was initiated in 1973 by Erdős and Simonovits. Recently $δ_χ(H)$ was determined for all graphs $H$. It is known that $δ_χ(H,p) =δ_χ(H)$ for all fixed $p \in (0,1)$, but that typically $δ_χ(H,p) \ne δ_χ(H)$ if $p = o(1)$. Here we study the problem for sparse random graphs. We determine $δ_χ(H,p)$ for most functions $p = p(n)$ when $H\in\{K_3,C_5\}$, and also for all graphs $H$ with $χ(H) \not\in \{3,4\}$.

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