Finding and using expanders in locally sparse graphs

We show that every locally sparse graph contains a linearly sized expanding subgraph. For constants $c_1>c_2>1$, $0<\alpha<1$, a graph $G$ on $n$ vertices is called a $(c_1,c_2,\alpha)$-graph if it has at least $c_1n$ edges, but every vertex subset $W\subset V(G)$ of size $|W|\le \alpha n$ spans less than $c_2|W|$ edges. We prove that every $(c_1,c_2,\alpha)$-graph with bounded degrees contains an induced expander on linearly many vertices. The proof can be made algorithmic. We then discuss several applications of our main result to random graphs, to problems about embedding graph minors, and to positional games.