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 Erdős 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) 6= 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) 6∈ {3, 4}.