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.