Extremal density for sparse minors and subdivisions.

We prove an asymptotically tight bound on the extremal density guaranteeing subdivisions of bounded-degree bipartite graphs with a mild separability condition. As corollaries, we answer several questions of Reed and Wood on embedding sparse minors. Among others, • average degree is sufficient to force the grid as a topological minor; • average degree forces every -vertex planar graph as a minor, and the constant is optimal, furthermore, surprisingly, the value is the same for -vertex graphs embeddable on any fixed surface; • a universal bound of on average degree forcing every -vertex graph in any nontrivial minor-closed family as a minor, and the constant 2 is best possible by considering graphs with given treewidth. [ABSTRACT FROM AUTHOR]