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Perfect stable regularity lemma and slice-wise stable hypergraphs

Artem Chernikov, Henry Towsner

TL;DR

The paper develops a comprehensive framework connecting stability notions for hypergraphs with measure-theoretic stable regularity lemmas, extending classical graph stability to 3-hypergraphs. It introduces partition-wise, slice-wise, and μ-stability, and shows how ultraproducts and graded probability spaces provide a bridge between finitary combinatorics and infinitary model theory. It proves a transfer principle linking perfect stable partitions in ultraproducts to approximately perfect partitions in finite graphs, and demonstrates both separations and coincidences among the various regularity notions, including a striking example of slice-wise stability without stable regularity. In the ternary setting, the authors classify the relationships among the stability variants, obtain strong stable regularity under certain directional slice-stability, and establish a slice-wise stable regularity lemma delivering 0/1 densities on a countable rectangle partition. The results advance tame hypergraph regularity theory, with implications for combinatorics and model theory via Keisler measures and ultraproduct methods, and open routes for characterizing stability-driven regular partitions in higher arities.

Abstract

We investigate various forms of (model-theoretic) stability for hypergraphs and their corresponding strengthenings of the hypergraph regularity lemma with respect to partitions of vertices. On the one hand, we provide a complete classification of the various possibilities in the ternary case. On the other hand, we provide an example of a family of slice-wise stable 3-hypergraphs so that for no partition of the vertices, any triple of parts has density close to 0 or 1. In particular, this addresses some questions and conjectures of Terry and Wolf. We work in the general measure theoretic context of graded probability spaces, so all our results apply both to measures in ultraproducts of finite graphs, leading to the aforementioned combinatorial applications, and to commuting definable Keisler measures, leading to applications in model theory.

Perfect stable regularity lemma and slice-wise stable hypergraphs

TL;DR

The paper develops a comprehensive framework connecting stability notions for hypergraphs with measure-theoretic stable regularity lemmas, extending classical graph stability to 3-hypergraphs. It introduces partition-wise, slice-wise, and μ-stability, and shows how ultraproducts and graded probability spaces provide a bridge between finitary combinatorics and infinitary model theory. It proves a transfer principle linking perfect stable partitions in ultraproducts to approximately perfect partitions in finite graphs, and demonstrates both separations and coincidences among the various regularity notions, including a striking example of slice-wise stability without stable regularity. In the ternary setting, the authors classify the relationships among the stability variants, obtain strong stable regularity under certain directional slice-stability, and establish a slice-wise stable regularity lemma delivering 0/1 densities on a countable rectangle partition. The results advance tame hypergraph regularity theory, with implications for combinatorics and model theory via Keisler measures and ultraproduct methods, and open routes for characterizing stability-driven regular partitions in higher arities.

Abstract

We investigate various forms of (model-theoretic) stability for hypergraphs and their corresponding strengthenings of the hypergraph regularity lemma with respect to partitions of vertices. On the one hand, we provide a complete classification of the various possibilities in the ternary case. On the other hand, we provide an example of a family of slice-wise stable 3-hypergraphs so that for no partition of the vertices, any triple of parts has density close to 0 or 1. In particular, this addresses some questions and conjectures of Terry and Wolf. We work in the general measure theoretic context of graded probability spaces, so all our results apply both to measures in ultraproducts of finite graphs, leading to the aforementioned combinatorial applications, and to commuting definable Keisler measures, leading to applications in model theory.
Paper Structure (23 sections, 41 theorems, 125 equations)

This paper contains 23 sections, 41 theorems, 125 equations.

Table of Contents

  1. Introduction
  2. Variants of stable regularity for hypergraphs
  3. History of the problem
  4. Summary of the results
  5. Choice of a setting and structure of the paper
  6. Acknowledgements
  7. Stability, partition-wise and slice-wise stability, measure stability
  8. Some notation
  9. Graded probability spaces
  10. Graded probability spaces as first-order structures and their ultraproducts
  11. Stability, trees and their measurable variants
  12. Perfect and decent sets and partitions, and the measure theoretic regularity lemma for $\mu$-stable hypergraphs
  13. Various versions of the stable regularity lemma for hypergraphs and a transfer principle
  14. Definitions
  15. Working Through an Example
  16. ...and 8 more sections

Key Result

Theorem A

Let $\mathcal{H}$ be a hereditarily closed family of finite $k$-partite $k$-hypergraphs. Then the following are equivalent:

Theorems & Definitions (145)

  • Definition 1.1
  • Definition 1.2
  • Theorem A
  • Theorem B
  • Theorem C: see Example
  • Theorem D
  • Theorem E
  • Theorem F
  • Theorem G
  • Definition 2.1
  • ...and 135 more