Pure Pairs. V. Excluding Some Long Subdivision.

Abstract

A \pure pair" in a graph G is a pair A;B of disjoint subsets of V (G) such that A is complete or anticomplete to B. Jacob Fox showed that for all " > 0, there is a comparability graph G with n vertices, where n is large, in which there is no pure pair A;B with jAj; jBj "n. He also proved that for all c > 0 there exists " > 0 such that for every comparability graph G with n > 1 vertices, there is a pure pair A;B with jAj; jBj "n1􀀀c; and conjectured that the same holds for every perfect graph G. We prove this conjecture and strengthen it in several ways. In particular, we show that for all c > 0, and all `1; `2 4=c + 9, there exists " > 0 such that, if G is an (n > 1)-vertex graph with no hole of length exactly `1 and no antihole of length exactly `2, then there is a pure pair A;B in G with jAj "n and jBj "n1􀀀c. This is further strengthened, replacing excluding a hole by excluding some \long" subdivision of a general graph.