Dense Minors from the Crux

Let $G$ be a graph with average degree $d>0$ and $C_α(G)$ be the minimum order of a subgraph whose average degree is at least $αd$ for any fixed $0<α<1$. Then $G$ has a minor of average degree $Ω(\sqrt{d\,C_α(G)})$, which is asymptotically tight up to a constant factor. This reveals a unified mechanism underlying previous results on dense minors in locally sparse graphs.