The extremal function for disconnected minors

For a graph $H$ let $c(H)$ denote the supremum of $|E(G)|/|V(G)|$ taken over all non-null graphs $G$ not containing $H$ as a minor. We show that $c(H) \leq \frac{|V(H)|+\mathrm{comp}(H)}{2}-1,$ 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.

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