Pure Pairs. VIII. Excluding a Sparse Graph

A pure pair of size t in a graph G is a pair A, B of disjoint subsets of V(G), each of cardinality at least t, such that A is either complete or anticomplete to B. It is known that, for every forest H, every graph on n≥2 vertices that does not contain H or its complement as an induced subgraph has a pure pair of size Ω(n); furthermore, this only holds when H or its complement is a forest. In this paper, we look at pure pairs of size n^1-c, where 0<c<1. Let H be a graph: does every graph on n≥2 vertices that does not contain H or its complement as an induced subgraph have a pure pair of size Ω(|G|^1-c)? The answer is related to the congestion of H, the maximum of 1-(|J|-1)/|E(J)| over all subgraphs J of H with an edge. (Congestion is nonnegative, and equals zero exactly when H is a forest.) Let d be the smaller of the congestions of H and H¯. We show that the answer to the question above is “yes” if d≤c/(9+15c), and “no” if d>c.