Induced subdivisions with pinned branch vertices

We prove that for all r∈N∪{0} and s,t∈N, there exists Ω=Ω(r,s,t)∈N with the following property. Let G be a graph and let H be a subgraph of G isomorphic to a (≤r)-subdivision of K_Ω. 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 (≤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.