Embedding clique subdivisions via crux

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