  We prove asymptotically optimal bounds on the number of edges a graph $G$
must have in order that any $r$-colouring of $E(G)$ has a colour class which
contains every $D$-degenerate graph on $n$ vertices with bounded maximum
degree. We also improve the upper bounds on the number of edges $G$ must have
in order that any $r$-colouring of $E(G)$ has a colour class which contains
every $n$-vertex graph with maximum degree $\Delta$, for each $\Delta\ge 4$. In
both cases, we show that a binomial random graph with $Cn$ vertices and a
suitable edge probability is likely to provide the desired $G$.
