Thomason's balanced clique-subdivision conjecture

A balanced subdivision of a graph $H$ is obtained by replacing every edge of $H$ with an internally vertex-disjoint path of the same length; write this as $TH^{(\ell)}$ when that length is $\ell$. For a graph $G$, let $\delta(G)$ denote its minimum degree. Thomason's conjecture. For every $k\ge 1$, there exists a function $f(k)$ such that if

$$\delta(G)\ge f(k),$$

then $G$ contains a balanced subdivision of $K_k$. This conjecture was confirmed by Liu and Montgomery, so the assertion is now a theorem; the paper studies stronger quantitative results for balanced clique subdivisions.