Minors in small-set expanders

We study large minors in small-set expanders. More precisely, we consider graphs with $n$ vertices and the property that every set of size at most $\alpha n / t$ expands by a factor of $t$, for some (constant) $\alpha > 0$ and large $t = t(n)$. We obtain the following: * Improving results of Krivelevich and Sudakov, we show that a small-set expander contains a complete minor of order $\sqrt{n t / \log n}$. * We show that a small-set expander contains every graph $H$ with $O(n \log t / \log n)$ edges and vertices as a minor. We complement this with an upper bound showing that if an $n$-vertex graph $G$ has average degree $d$, then there exists a graph with $O(n \log d / \log n)$ edges and vertices which is not a minor of $G$. This has two consequences: (i) It implies the optimality of our result in the case $t = d^c$ for some constant $c > 0$, and (ii) it shows expanders are optimal minor-universal graphs of a given average degree.