$L_p$ regular sparse hypergraphs: box norms

TL;DR

This paper explores variants of Gowers box norms and their application to sparse hypergraphs, establishing generalized theorems and examples of pseudorandom hypergraph families with key combinatorial properties.

Contribution

It introduces generalized von Neumann theorems for $L_p$ graphons and provides examples of pseudorandom hypergraph families satisfying counting and removal lemmas.

Findings

Proved a generalized von Neumann theorem for $L_p$ graphons

Constructed examples of pseudorandom sparse hypergraphs

Established the relevance of box norms in sparse hypergraph analysis

Abstract

We consider some variants of the Gowers box norms, introduced by Hatami, and show their relevance in the context of sparse hypergraphs. Our main results are the following. Firstly, we prove a generalized von Neumann theorem for $L_p$ graphons. Secondly, we give natural examples of pseudorandom families, that is, sparse weighted uniform hypergraphs which satisfy relative versions of the counting and removal lemmas.