Minors in small-set expanders

We study large minors in small-set expanders. More precisely, we consider graphs with n n vertices and the property that every set of size at most α n / t \alpha n / t expands by a factor of t t , for some (constant) α > 0 \alpha > 0 and large t = t ( n ) 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 n t / log ⁡ n \sqrt {n t / \log n} . We show that a small-set expander contains every graph H H with O ( n log ⁡ t / log ⁡ n ) O(n \log t / \log n) edges and vertices as a minor. We complement this with an upper bound showing that if an n n -vertex graph G G has average degree d d , then there exists a graph with O ( n log ⁡ d / log ⁡ n ) O(n \log d / \log n) edges and vertices which is not a minor of