Chromatic thresholds in dense random graphs

Abstract

The chromatic threshold δχ(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⊂G(n,p) with minimum degree δ(G)⩾dpn has bounded chromatic number. The study of the parameter δχ(H):=δχ(H,1) was initiated in 1973 by Erdős and Simonovits, and was recently determined for all graphs H. In this paper we show that δχ(H,p)=δχ(H) for all fixed p∈(0,1), but that typically δχ(H,p)≠δχ(H) if p=o(1). We also make significant progress towards determining δχ(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. © 2017 Wiley Periodicals, Inc. Random Struct. Alg., 51, 185–214, 2017