$L_p$ regular sparse hypergraphs

TL;DR

This paper investigates sparse hypergraphs with $L_p$ regularity, establishing regularity and counting lemmas, and extending the relative removal lemma to this setting, thus addressing a question posed by prior researchers.

Contribution

It introduces $L_p$ regularity for sparse hypergraphs and extends key combinatorial lemmas to this framework, advancing understanding of pseudorandom hypergraph structures.

Findings

Proved regularity and counting lemmas for $L_p$ regular hypergraphs.

Extended the relative removal lemma to the $L_p$ regular setting.

Provided answers to a question posed by Borgs, Chayes, Cohn, and Zhao.

Abstract

We study sparse hypergraphs which satisfy a mild pseudorandomness condition known as $L_p$ regularity. We prove appropriate regularity and counting lemmas, and we extend the relative removal lemma of Tao in this setting. This answers a question of Borgs, Chayes, Cohn and Zhao.