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,
β(1+o(1))t2 average degree is sufficient to force the tΓt
grid as a topological minor;
β(3/2+o(1))t average degree forces every t-vertex planar graph
as a minor, and the constant 3/2 is optimal, furthermore, surprisingly, the
value is the same for t-vertex graphs embeddable on any fixed surface;
β a universal bound of (2+o(1))t on average degree forcing every
t-vertex graph in any nontrivial minor-closed family as a minor, and the
constant 2 is best possible by considering graphs with given treewidth.Comment: 33 pages, 6 figure