  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
the parameter $\delta_\chi(H) := \delta_\chi(H,1)$ 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 $\delta_\chi(H,p) = \delta_\chi(H)$ for all fixed $p
\in (0,1)$, but that typically $\delta_\chi(H,p) \ne \delta_\chi(H)$ if $p =
o(1)$. We also make significant progress towards determining $\delta_\chi(H,p)$
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.
