Pure Pairs. II. Excluding All Subdivisions of A Graph

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