Pure Pairs. II. Excluding All Subdivisions of A Graph

We prove for every graph H there exists ɛ > 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 ɛ| G | neighbours, or there are two disjoint sets A , B ⊆ V ( G ) with | A |,| B |≥ɛ| 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.

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