Pseudofinite proofs of the stable graph regularity lemma

TL;DR

This paper presents a detailed pseudofinite proof of the stable graph regularity lemma, leveraging local stability theory and ultraproducts, and introduces a qualitative strengthening through the concept of functional error.

Contribution

It provides a simplified, detailed pseudofinite proof of the stable graph regularity lemma and demonstrates how functional error can enhance partition extraction methods.

Findings

Simplified proof of the stable graph regularity lemma using pseudofinite methods

Introduction of a qualitative strengthening via functional error

Elementary argument for extracting equipartitions from arbitrary partitions

Abstract

This expository article is based on two lectures given by the first author at the Fields Institute in the Fall 2021 Thematic Program on Trends in Pure and Applied Model Theory. We give a detailed proof of a qualitative version of the Mallaris-Shelah regularity lemma for stable graphs using only basic local stability theory and an ultraproduct construction. This proof strategy was first established by Malliaris and Pillay, and later simplified by Pillay. We provide some further simplifications, and also explain how the pseudofinite approach can be used to obtain a qualitative strengthening (compared to previous proofs) in terms of "functional error". To illustrate the extra leverage obtained by functional error, we give an elementary argument for extracting equipartitions from arbitrary partitions.