The Quasi-Randomness of Hypergraph Cut Properties

TL;DR

This paper investigates whether certain hypergraph cut properties imply quasi-randomness, revealing that only a specific symmetric case does not guarantee quasi-randomness and characterizing the unique non-quasi-random hypergraph with this property.

Contribution

It resolves a 1991 open problem by showing the conditions under which hypergraph cut properties imply quasi-randomness and characterizes the unique non-quasi-random hypergraph satisfying these properties.

Findings

Hypergraph cut properties do not imply quasi-randomness unless the partition is symmetric.

The only non-quasi-random hypergraph satisfying the property is uniquely characterized.

The proof combines probabilistic, algebraic, and association scheme techniques.

Abstract

Let a_1,...,a_k satisfy a_1+...+a_k=1 and suppose a k-uniform hypergraph on n vertices satisfies the following property; in any partition of its vertices into k sets A_1,...,A_k of sizes a_1*n,...,a_k*n, the number of edges intersecting A_1,...,A_k is the number one would expect to find in a random k-uniform hypergraph. Can we then infer that H is quasi-random? We show that the answer is negative if and only if a_1=...=a_k=1/k. This resolves an open problem raised in 1991 by Chung and Graham [J. AMS '91]. While hypergraphs satisfying the property corresponding to a_1=...=a_k=1/k are not necessarily quasi-random, we manage to find a characterization of the hypergraphs satisfying this property. Somewhat surprisingly, it turns out that (essentially) there is a unique non quasi-random hypergraph satisfying this property. The proofs combine probabilistic and algebraic arguments with results from the theory of association schemes.