Induced trees in graphs of large chromatic number

Gyarfas and Sumner independently conjectured that for every tree T and integer k there is an integer f(k, T ) such that every graph G with χ(G) > f(k, T ) contains either Kk or an induced copy of T . We prove a ‘topological’ version of the conjecture: for every tree T and integer k there is g(k, T ) such that every graph G with χ(G) > g(k, T ) contains either Kk or an induced copy of a subdivision of T .