Natural quasirandomness properties

TL;DR

This paper introduces three new, natural hierarchies of quasirandomness properties applicable to various combinatorial objects, providing a unified framework with multiple characterizations and relationships to existing hypergraph quasirandomness concepts.

Contribution

It proposes a general, formal approach to quasirandomness for arbitrary combinatorial objects, extending beyond case-by-case methods and establishing equivalences and separations among the properties.

Findings

New hierarchies of quasirandomness properties for combinatorial objects

Equivalent characterizations similar to hypergraph quasirandomness

Implications and separations among the proposed properties

Abstract

The theory of quasirandomness has greatly expanded from its inaugural graph theoretical setting to several different combinatorial objects such as hypergraphs, tournaments, permutations, etc. However, these quasirandomness variants have been done in an ad-hoc case-by-case manner. In this paper, we propose three new hierarchies of quasirandomness properties that can be naturally defined for arbitrary combinatorial objects. Our properties are also "natural" in more formal sense: they are preserved by local combinatorial constructions (encoded by open interpretations). We show that our quasirandomness properties have several different but equivalent characterizations that are similar to hypergraph quasirandomness properties. We also prove several implications and separations comparing them to each other and to what has been known for hypergraphs. The main notion explored by our statements and proofs is that of unique coupleability: two limit objects are uniquely coupleable if there is a unique limit object in the combined theory that is an alignment (i.e., a coupling) of these two objects.