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 $\varepsilon >0$, there is a comparability graph G with n vertices, where n is large, in which there is no pure pair A, B with $|A|,|B|\ge \varepsilon n$. He also proved that for all $c>0$ there exists $\varepsilon >0$ such that for every comparability graph G with $n>1$ vertices, there is a pure pair A, B with $|A|,|B|\ge \varepsilon n^{1-c}$; 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 $\ell _1, \ell _2\ge 4/c+9$, there exists $\varepsilon >0$ such that, if G is an $(n>1)$-vertex graph with no hole of length exactly $\ell _1$ and no antihole of length exactly $\ell _2$, then there is a pure pair A, B in G with $|A|\ge \varepsilon n$ and $|B|\ge \varepsilon n^{1-c}$. This is further strengthened, replacing excluding a hole by excluding some “long” subdivision of a general graph.