Pure pairs. II. Excluding all subdivisions of a graph

TL;DR

The paper proves a structural property of graphs avoiding subdivisions of a fixed graph H, showing such graphs must contain large homogeneous sets or vertices with many neighbors, relating to the Erdős-Hajnal conjecture.

Contribution

It establishes a new extremal result linking subdivisions of a graph H to large homogeneous sets in graphs, advancing understanding of the Erdős-Hajnal conjecture.

Findings

Graphs avoiding subdivisions of H have vertices with many neighbors or large bipartite structures.

Such graphs or their complements contain large cliques or stable sets of polynomial size.

The results connect subdivision avoidance to the Erdős-Hajnal property.

Abstract

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