The extremal function for disconnected minors

For a graph $H$ let $c(H)$ denote the supremum of $\mathrm{\mid }E(G)\mathrm{\mid }\mathrm{/}\mathrm{\mid }V(G)\mathrm{\mid }$ taken over all non-null graphs $G$ not containing $H$ as a minor. We show that $$c(H)\le \frac{\mathrm{\mid }V(H)\mathrm{\mid }+\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}(H)}{2}-1,$$ when $H$ is a union of cycles, verifying conjectures of Reed and Wood, and Harvey and Wood. We derive the above result from a theorem which allows us to find two vertex disjoint subgraphs with prescribed densities in a sufficiently dense graph, which might be of independent interest.