Logarithmically small minors and topological minors

For every integer t there is a smallest real number c(t) such that any graph with average degree at least c(t) must contain a Kt-minor (proved by Mader). Improving on results of Shapira and Sudakov, we prove the conjecture of Fiorini, Joret, Theis and Wood that any graph with n vertices and average degree at least c(t) + ε must contain a Kt-minor consisting of at most C(ε, t) logn vertices. Mader also proved that for every integer t there is a smallest real number s(t) such that any graph with average degree larger than s(t) must contain a Kttopological minor. We prove that, for sufficiently large t, graphs with average degree at least (1 + ε)s(t) contain a Kt-topological minor consisting of at most C(ε, t) logn vertices. Finally, we show that, for sufficiently large t, graphs with average degree at least (1 + ε)c(t) contain either a Kt-minor consisting of at most C(ε, t) vertices or a Kt-topological minor consisting of at most C(ε, t) logn vertices.