The Poset of Hypergraph Quasirandomness

TL;DR

This paper maps out the implication relationships among various hypergraph quasirandom properties, clarifying how different notions of quasirandomness relate to each other in hypergraphs.

Contribution

It determines the poset of implications between hypergraph quasirandom properties, resolving a long-standing open problem in the field.

Findings

Established the partial order of implications among hypergraph quasirandom properties

Resolved a recent open question by Chung about the structure of these properties

Extended the foundational work of Chung and Graham from the 1990s

Abstract

Chung and Graham began the systematic study of k-uniform hypergraph quasirandom properties soon after the foundational results of Thomason and Chung-Graham-Wilson on quasirandom graphs. One feature that became apparent in the early work on k-uniform hypergraph quasirandomness is that properties that are equivalent for graphs are not equivalent for hypergraphs, and thus hypergraphs enjoy a variety of inequivalent quasirandom properties. In the past two decades, there has been an intensive study of these disparate notions of quasirandomness for hypergraphs, and an open problem that has emerged is to determine the relationship between them. Our main result is to determine the poset of implications between these quasirandom properties. This answers a recent question of Chung and continues a project begun by Chung and Graham in their first paper on hypergraph quasirandomness in the early 1990's.