Counting results for sparse pseudorandom hypergraphs I

TL;DR

This paper proves a counting lemma for embedding linear uniform hypergraphs into sparse pseudorandom hypergraphs, extending previous graph results to hypergraphs and enabling further applications in the field.

Contribution

It introduces a new counting lemma for hypergraphs that generalizes earlier graph results, facilitating embeddings in sparse pseudorandom hypergraphs.

Findings

Established a counting lemma for hypergraph embeddings

Generalized graph embedding results to hypergraphs

Enabled new applications in hypergraph theory

Abstract

We establish a so-called counting lemma that allows embeddings of certain linear uniform hypergraphs into sparse pseudorandom hypergraphs, generalizing a result for graphs [Embedding graphs with bounded degree in sparse pseudorandom graphs, Israel J. Math. 139 (2004), 93-137]. Applications of our result are presented in the companion paper [Counting results for sparse pseudorandom hypergraphs II].