Sparse graphs without linear anticomplete pairs

We prove for every graph H there exists > 0 such that, for every graph G with |G| ≥ 2, if no induced subgraph of G is a subdivision of H, then either some vertex of G has at least |G| neighbours, or there are two disjoint sets A,B ⊆ V (G) with |A|, |B| ≥ |G| such that no edge joins A and B. It follows that for every graph H, there exists c > 0 such that for every graph G, if no induced subgraph of G or its complement is a subdivision of H, then G has a clique or stable set of cardinality at least |G|c. This is related to the Erdős-Hajnal conjecture.