Averages of hypergraphs and higher arity stability

TL;DR

This paper establishes a strong hypergraph regularity property for multi-parameter set families and connects it to higher arity stability in model theory, showing that certain complex hypergraphs still satisfy regularity despite stability failures.

Contribution

It introduces a new regularity lemma for hypergraphs related to higher arity stability, expanding understanding of stability phenomena in model theory and combinatorics.

Findings

Hypergraph averages satisfy a strong regularity property.

Certain hypergraphs embedding into specific structures still obey regularity lemmas.

Strong ternary stability cannot be characterized solely by excluded hypergraphs.

Abstract

We show that $k$-ary functions giving the measure of the intersection of multi-parametric families of sets in probability spaces, e.g. $(x,y,z) \in X \times Y \times Z \mapsto \mu(P_{x,y} \cap Q_{x,z} \cap R_{y,z})$, satisfy a particularly strong form of hypergraph regularity. More generally, this applies to the (integral) averages of continuous combinations of functions of smaller arity. This result is connected to higher arity stability in model theory, that we discuss in the second part of the paper. We demonstrate that all hypergraphs embedding both into the half-simplex and into $GS(\mathbb{F}_3)$, the two known sources of failure of ternary stability, do satisfy an analogous regularity lemma -- hence strong ternary stability cannot be characterized simply by excluded hypergraphs.