Induced subdivisions with pinned branch vertices

We prove that for all $r\in N\cup \left\{0\right\}$ and $s,t\in N$, there exists $\Omega =\Omega (r,s,t)\in N$ with the following property. Let $G$ be a graph and let $H$ be a subgraph of $G$ isomorphic to a $(\le r)$-subdivision of $K_{\Omega }$. Then either $G$ contains $K_t$ or $K_{t,t}$ as an induced subgraph, or there is an induced subgraph $J$ of $G$ isomorphic to a proper $(\le r)$-subdivision of $K_s$ such that every branch vertex of $J$ is a branch vertex of $H$. This answers in the affirmative a question of Lozin and Razgon. In fact, we show that both the branch vertices and the paths corresponding to the subdivided edges between them can be preserved.