Szemer\'edi's regularity lemma via martingales

TL;DR

This paper presents a new proof of a generalized Szemerédi's regularity lemma using martingale difference sequences, applicable to various structures and $L_p$ spaces, expanding its theoretical framework.

Contribution

It introduces a martingale-based approach to a probabilistic version of Szemerédi's regularity lemma, broadening its applicability to multiple structures and $L_p$ spaces.

Findings

Proves a variant of the probabilistic Szemerédi's regularity lemma.

Applies to graphs, hypergraphs, hypercubes, and graphons.

Uses martingale difference sequences for the proof.

Abstract

We prove a variant of the abstract probabilistic version of Szemer\'edi's regularity lemma, due to Tao, which applies to a number of structures (including graphs, hypergraphs, hypercubes, graphons, and many more) and works for random variables in $L_p$ for any $p>1$. Our approach is based on martingale difference sequences.