Partition universality for graphs of bounded degeneracy and degree

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 $G$.