Chromatic thresholds in dense random graphs

Abstract

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 the parameter $δ_χ(H) := δ_χ(H,1)$ was initiated in 1973 by Erdős and Simonovits, and was recently determined for all graphs $H$. In this paper we show 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 $p = o(1)$. We also make significant progress towards determining $δ_χ(H,p)$ for all graphs $H$ in the range $p = n^{-o(1)}$. In sparser random graphs the problem is somewhat more complicated, and is studied in a separate paper.

36 pages (including appendix), 1 figure; the appendix is copied with minor modifications from arXiv:1108.1746 for a self-contained proof of a technical lemma; accepted to Random Structures and Algorithms