Chromatic thresholds in sparse random graphs

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 δχ(H):=δχ(H,1) was initiated in 1973 by Erd\H{o}s and Simonovits. Recently δχ(H) was determined for all graphs H. It is known that δχ(H,p)=δχ(H) for all fixed p∈(0,1), but that typically δχ(H,p)≠δχ(H) if p=o(1). Here we study the problem for sparse random graphs. We determine δχ(H,p) for most functions p=p(n) when H∈{K3,C5}, and also for all graphs H with χ(H)∉{3,4}.