Quasirandomness in hypergraphs

TL;DR

This paper explores various notions of quasirandomness in hypergraphs, providing combinatorial proofs of key equivalences that were previously established through analytic methods.

Contribution

It offers short, purely combinatorial proofs of the main equivalences of hypergraph quasirandomness notions, simplifying previous analytic approaches.

Findings

Provided combinatorial proofs of hypergraph quasirandomness equivalences

Extended the understanding of quasirandom properties beyond graphs

Simplified the proof techniques for hypergraph quasirandomness

Abstract

An $n$-vertex graph $G$ of edge density $p$ is considered to be quasirandom if it shares several important properties with the random graph $G(n,p)$. A well-known theorem of Chung, Graham and Wilson states that many such `typical' properties are asymptotically equivalent and, thus, a graph $G$ possessing one such property automatically satisfies the others. In recent years, work in this area has focused on uncovering more quasirandom graph properties and on extending the known results to other discrete structures. In the context of hypergraphs, however, one may consider several different notions of quasirandomness. A complete description of these notions has been provided recently by Towsner, who proved several central equivalences using an analytic framework. We give short and purely combinatorial proofs of the main equivalences in Towsner's result.