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Classification properties for some ternary structures

Alberto Miguel-Gómez

TL;DR

The paper identifies the first natural example of a strictly NSOP$_4$ structure with no invariant independence relation over models satisfying full existence and stationarity. It proves, via two distinct routes, that the countable homogeneous $ extnormal{H}_4$-free 3-hypertournament has $ extnormal{SOP}_3$, $ extnormal{TP}_2$, and $ extnormal{NSOP}_4$, while also exhibiting $ extnormal{IP}_2$ and $ extnormal{NFOP}_3$. The first proof relies on concrete combinatorial constructions to establish SOP$_3$, IP$_2$, and TP$_2$, implying NSOP$_4$, whereas the second proof adapts Mutchnik’s framework by introducing non-stationary independence notions (notably leq$_K$ and hti) and demonstrating a relative Kim lemma in this non-stationary setting. Collectively, the results illuminate how higher-arity generalizations of stability theory interact with invariant independence notions in a natural combinatorial structure, potentially guiding future NSOP$_4$ classifications and independence frameworks. The findings have significance for higher-arity stability theory and structural Ramsey phenomena related to hypertournament classes.

Abstract

We provide a model-theoretic classification of the countable homogeneous $\mathbf{H}_4$-free 3-hypertournament studied by Cherlin, Hubička, Konečný, and Nešetřil. Our main result is that the theory of this structure is $\mathrm{SOP}_3$, $\mathrm{TP}_2$, and $\mathrm{NSOP}_4$. We offer two proofs of this fact: one is a direct proof, and the other employs part of the abstract machinery recently developed by Mutchnik.

Classification properties for some ternary structures

TL;DR

The paper identifies the first natural example of a strictly NSOP structure with no invariant independence relation over models satisfying full existence and stationarity. It proves, via two distinct routes, that the countable homogeneous -free 3-hypertournament has , , and , while also exhibiting and . The first proof relies on concrete combinatorial constructions to establish SOP, IP, and TP, implying NSOP, whereas the second proof adapts Mutchnik’s framework by introducing non-stationary independence notions (notably leq and hti) and demonstrating a relative Kim lemma in this non-stationary setting. Collectively, the results illuminate how higher-arity generalizations of stability theory interact with invariant independence notions in a natural combinatorial structure, potentially guiding future NSOP classifications and independence frameworks. The findings have significance for higher-arity stability theory and structural Ramsey phenomena related to hypertournament classes.

Abstract

We provide a model-theoretic classification of the countable homogeneous -free 3-hypertournament studied by Cherlin, Hubička, Konečný, and Nešetřil. Our main result is that the theory of this structure is , , and . We offer two proofs of this fact: one is a direct proof, and the other employs part of the abstract machinery recently developed by Mutchnik.
Paper Structure (16 sections, 27 theorems, 9 equations)

This paper contains 16 sections, 27 theorems, 9 equations.

Table of Contents

  1. Introduction
  2. Conventions and preliminaries
  3. Fraïssé's Theorem
  4. Generalised stability theory
  5. A primer on independence relations
  6. Relative Kim independence
  7. Basic properties
  8. Definition of the H4-free 3-hypertournament
  9. Existence of invariant extensions
  10. First proof of NSOP4
  11. leqK-independence
  12. Defining leqK-independence
  13. A criterion for the strong witnessing property
  14. Second proof of NSOP4
  15. Triviality of Conant-independence
  16. ...and 1 more sections

Key Result

Theorem 1.0.1

The theory of the countable homogeneous $\mathbf{H}_4$-free 3-hypertournament is $\textnormal{SOP}_3$, $\textnormal{TP}_2$, and $\textnormal{NSOP}_4$.

Theorems & Definitions (96)

  • Theorem 1.0.1
  • Theorem 1.0.2
  • Definition 2.1.1
  • Definition 2.1.2
  • Definition 2.2.1
  • Definition 2.2.2
  • Example 2.3.1
  • Definition 2.3.2
  • Example 2.3.4
  • Definition 2.3.5
  • ...and 86 more