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Irregular triads in 3-uniform hypergraphs

C. Terry, J. Wolf

TL;DR

This work extends the graph-theoretic regularity paradigm to 3-uniform hypergraphs, introducing irregular triads and higher-arity tameness notions such as disc_{2,3}-regularity, vdisc_3-regularity, and linear disc_{2,3}-error. It establishes a sharp dichotomy: disc_{2,3}-homogeneity is equivalent to bounded VC_2-dimension, and the NFOP_2 class corresponds precisely to linear disc_{2,3}-error, with strong regularity tools and slicewise stability guiding the structure. The paper also develops encodings of higher-order order properties (FOP_2/IP_2) and proves a robust strong-stability regularity lemma, along with removal-type results, to support the linear-error theory. Together, these results map out a multifaceted landscape of tameness versus complexity in 3-graphs, with implications for hypergraph regularity, counting, and model-theoretic connections in finite combinatorics.

Abstract

Over the past several years, numerous authors have explored model theoretically motivated combinatorial conditions that ensure that a graph has an efficient regular decomposition in the sense of Szemerédi. In this paper we set out a research program that explores a corresponding set of questions for 3-uniform hypergraphs, a setting in which useful notions of regularity are significantly more intricate. The main results in this paper concern certain combinatorial properties which arose as natural higher-order generalizations of the order property in parallel work of the authors in the arithmetic setting. Interpreted in the context of 3-uniform hypergraphs, these are tightly connected to the nature of irregular triads. Specifically, we show that a hereditary property of 3-uniform hypergraphs admits regular decompositions with so-called "linear error" if and only if it does not have the functional order property. Along the way, we show that a hereditary property of 3-uniform hypergraphs is homogeneous (i.e. all regular triads have density near $0$ or near $1$) if and only it has bounded $\textrm{VC}_2$-dimension. This provides a quantitative version of a recent result of Chernikov and Towsner. We also address several questions arising from prior work on tame regularity in hypergraphs. In particular, we characterize the hereditary properties of $3$-uniform hypergraphs admitting the type of regular partitions appearing in work of Fox et al. as those that have bounded slicewise $\textrm{VC}$-dimension. This is again analogous to a recent non-quantitative result of Chernikov and Towsner.

Irregular triads in 3-uniform hypergraphs

TL;DR

This work extends the graph-theoretic regularity paradigm to 3-uniform hypergraphs, introducing irregular triads and higher-arity tameness notions such as disc_{2,3}-regularity, vdisc_3-regularity, and linear disc_{2,3}-error. It establishes a sharp dichotomy: disc_{2,3}-homogeneity is equivalent to bounded VC_2-dimension, and the NFOP_2 class corresponds precisely to linear disc_{2,3}-error, with strong regularity tools and slicewise stability guiding the structure. The paper also develops encodings of higher-order order properties (FOP_2/IP_2) and proves a robust strong-stability regularity lemma, along with removal-type results, to support the linear-error theory. Together, these results map out a multifaceted landscape of tameness versus complexity in 3-graphs, with implications for hypergraph regularity, counting, and model-theoretic connections in finite combinatorics.

Abstract

Over the past several years, numerous authors have explored model theoretically motivated combinatorial conditions that ensure that a graph has an efficient regular decomposition in the sense of Szemerédi. In this paper we set out a research program that explores a corresponding set of questions for 3-uniform hypergraphs, a setting in which useful notions of regularity are significantly more intricate. The main results in this paper concern certain combinatorial properties which arose as natural higher-order generalizations of the order property in parallel work of the authors in the arithmetic setting. Interpreted in the context of 3-uniform hypergraphs, these are tightly connected to the nature of irregular triads. Specifically, we show that a hereditary property of 3-uniform hypergraphs admits regular decompositions with so-called "linear error" if and only if it does not have the functional order property. Along the way, we show that a hereditary property of 3-uniform hypergraphs is homogeneous (i.e. all regular triads have density near or near ) if and only it has bounded -dimension. This provides a quantitative version of a recent result of Chernikov and Towsner. We also address several questions arising from prior work on tame regularity in hypergraphs. In particular, we characterize the hereditary properties of -uniform hypergraphs admitting the type of regular partitions appearing in work of Fox et al. as those that have bounded slicewise -dimension. This is again analogous to a recent non-quantitative result of Chernikov and Towsner.

Paper Structure

This paper contains 35 sections, 95 theorems, 539 equations.

Table of Contents

  1. Introduction
  2. Formal statements of problems and results
  3. Quasirandomness and regularity in $3$-graphs
  4. Statements of problems
  5. Statements of results
  6. Outline of paper
  7. Summary of problems and updates
  8. Preliminaries
  9. Notions of quasirandomness for graphs and hypergraphs
  10. Lemmas for slicing, intersecting, and combining decompositions
  11. Ramsey facts
  12. $\textnormal{VC}_2$-dimension and $\textnormal{disc}_{2,3}$-homogeneity
  13. Slicewise NIP, slicewise stability, and $\textnormal{vdisc}_3$-homogeneity
  14. The $\sim$-classes of slicewise stable and slicewise NIP properties
  15. Slicewise $\textnormal{VC}$-dimension and $\textnormal{vdisc}_3$-homogeneity
  16. ...and 20 more sections

Key Result

Theorem 1.1

Suppose $\mathcal{G}$ is a hereditary graph property. Then one of the following holds.

Theorems & Definitions (237)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.7
  • Definition 2.1: $\textnormal{disc}_2$
  • Definition 2.2: $\textnormal{vdisc}_3$
  • Definition 2.3: $\textnormal{vdisc}_3$-regular triples and partitions
  • Theorem 2.4: $\textnormal{vdisc}_3$-regularity lemma for 3-graphs, Chung.1991
  • Definition 2.5: $\textnormal{disc}_{2,3}$
  • Definition 2.6: $(t,\ell,\epsilon_1,\epsilon_2)$-decomposition
  • Definition 2.7: $\textnormal{disc}_{2,3}$-regular triads and decompositions
  • ...and 227 more