Large Complete Minors from a Cheeger Condition

Let $G$ be a graph on $n$ vertices, and let $e_G(S,V(G)\setminus S)$ be the number of edges with exactly one endpoint in $S$. The Cheeger constant and the restricted Cheeger constant of $G$, where $k\ge1$ is real, are, respectively, \[ h(G)=\min_{\substack{\emptyset\ne S\subseteq V(G)\\|S|\le \fra