  A graph $G$ is $q$-Ramsey for another graph $H$ if in any $q$-edge-colouring
of $G$ there is a monochromatic copy of $H$, and the classic Ramsey problem
asks for the minimum number of vertices in such a graph. This was broadened in
the seminal work of Burr, Erd\H{o}s, and Lov\'asz to the investigation of other
extremal parameters of Ramsey graphs, including the minimum degree.
  It is not hard to see that if $G$ is minimally $q$-Ramsey for $H$ we must
have $\delta(G) \ge q(\delta(H) - 1) + 1$, and we say that a graph $H$ is
$q$-Ramsey simple if this bound can be attained. Grinshpun showed that this is
typical of rather sparse graphs, proving that the random graph $G(n,p)$ is
almost surely $2$-Ramsey simple when $\frac{\log n}{n} \ll p \ll n^{-2/3}$. In
this paper, we explore this question further, asking for which pairs $p = p(n)$
and $q = q(n,p)$ we can expect $G(n,p)$ to be $q$-Ramsey simple. We resolve the
problem for a wide range of values of $p$ and $q$; in particular, we uncover
some interesting behaviour when $n^{-2/3} \ll p \ll n^{-1/2}$.
