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 αn/t expands by a factor of t, for some (constant) α > 0 and large t = t(n). We obtain the following: • 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.