For a graph $G$ and a constant $α>0$, we denote by $C_α(G)$ the minimum order of a subgraph $H\subseteq G$ with $d(H)\ge αd(G)$. Liu and Montgomery conjectured that every graph $G$ contains $K_{Ω(t)}$ as a subdivision for $t=\min \{d(G), \sqrt{\tfrac{C_α(G)}{\log C_α(G)}}\}$. In the paper, we prove this conjecture.