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
Erd\H{o}s 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.Comment: 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 Algorithm
