  The chromatic threshold $\delta_\chi(H,p)$ 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 $\delta(G) \ge dpn$ has bounded chromatic number. The study of
$\delta_\chi(H) :=\delta_\chi(H,1)$ was initiated in 1973 by Erd\H{o}s and
Simonovits. Recently $\delta_\chi(H)$ was determined for all graphs $H$. It is
known that $\delta_\chi(H,p) =\delta_\chi(H)$ for all fixed $p \in (0,1)$, but
that typically $\delta_\chi(H,p) \ne \delta_\chi(H)$ if $p = o(1)$.
  Here we study the problem for sparse random graphs. We determine
$\delta_\chi(H,p)$ for most functions $p = p(n)$ when $H\in\{K_3,C_5\}$, and
also for all graphs $H$ with $\chi(H) \not\in \{3,4\}$.
