Szemer\'edi's regularity lemma revisited

TL;DR

This paper reexamines Szemerédi's regularity lemma through probability and information theory, introducing a new parameter and a stronger variant that extends to hypergraphs, offering a novel perspective on a fundamental combinatorial tool.

Contribution

It presents a new probabilistic and information-theoretic perspective on Szemerédi's regularity lemma, including a stronger version with a new parameter, applicable to hypergraphs.

Findings

Introduces a variant of Szemerédi's regularity lemma with a new parameter F.

Reproves the regularity lemma for hypergraphs using the new approach.

Provides insights into the lemma's structure from probability and information theory.

Abstract

Szemer\'edi's regularity lemma is a basic tool in graph theory, and also plays an important role in additive combinatorics, most notably in proving Szemer\'edi's theorem on arithmetic progressions . In this note we revisit this lemma from the perspective of probability theory and information theory instead of graph theory, and observe a variant of this lemma which introduces a new parameter $F$. This stronger version of the regularity lemma was iterated in a recent paper of the author to reprove the analogous regularity lemma for hypergraphs.