$L_p$ regular sparse hypergraphs: box norms
Volume 248 / 2020
Fundamenta Mathematicae 248 (2020), 49-77
MSC: Primary 05C35, 05C65; Secondary 46B20, 46B25.
DOI: 10.4064/fm656-10-2018
Published online: 17 June 2019
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.
