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)≤[Forumla presented]−1, when H is a union of cycles. This verifies a conjecture of Reed and Wood, and another conjecture of 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. © 2017 Elsevier Inc.