Quasi-random graphs and graph limits

TL;DR

This paper employs graph limit theory to analyze quasi-random properties of graphs, demonstrating that certain hereditary subgraph count properties are characterized by constant graphons, offering cleaner proofs than traditional methods.

Contribution

It introduces a graph limit approach to quasi-random properties, avoiding complex regularity lemma arguments and providing new measure-theoretic insights.

Findings

Quasi-random properties correspond to constant graphons.

Graph limit approach simplifies proofs of quasi-randomness.

Measure-theoretic issues are addressed in the analysis.

Abstract

We use the theory of graph limits to study several quasi-random properties, mainly dealing with various versions of hereditary subgraph counts. The main idea is to transfer the properties of (sequences of) graphs to properties of graphons, and to show that the resulting graphon properties only can be satisfied by constant graphons. These quasi-random properties have been studied before by other authors, but our approach gives proofs that we find cleaner, and which avoid the error terms and epsilons in the traditional arguments using the Szemeredi regularity lemma. On the other hand, other technical problems sometimes arise in analysing the graphon properties; in particular, a measure-theoretic problem on elimination of null sets that arises in this way is treated in an appendix.