Pure pairs. V. Excluding some long subdivision

TL;DR

This paper proves a conjecture that perfect graphs excluding certain long subdivisions always contain large pure pairs, extending previous results from comparability graphs and broadening the class of graphs where such pairs exist.

Contribution

It confirms and strengthens the conjecture that perfect graphs excluding specific long subdivisions have large pure pairs, generalizing prior results for comparability graphs.

Findings

Proved the conjecture for perfect graphs excluding certain long subdivisions.

Established existence of large pure pairs in graphs with no specific long holes or antiholes.

Extended results to graphs excluding some long subdivisions of general graphs.

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 $\epsilon>0$, there is a comparability graph $G$ with $n$ vertices, where $n$ is large, in which there is no pure pair $A,B$ with $|A|,|B|\ge \epsilon n$. He also proved that for all $c>0$ there exists $\epsilon>0$ such that for every comparability graph $G$ with $n>1$ vertices, there is a pure pair $A,B$ with $|A|,|B|\ge \epsilon n^{1-c}$; 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 $\ell_1, \ell_2\ge 4c^{-1}+9$, there exists $\epsilon>0$ such that, if $G$ is a graph with $n>1$ vertices and 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 $|A|\ge \epsilon n$ and $|B|\ge \epsilon n^{1-c}$. This is further strengthened, replacing excluding a hole by excluding some long subdivision of a general graph.