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Ramsey expansions of metrically homogeneous graphs

Andrés Aranda, David Bradley-Williams, Jan Hubička, Miltiadis Karamanlis, Michael Kompatscher, Matěj Konečný, Micheal Pawliuk

TL;DR

This work provides a near-complete characterization of Ramsey expansions, coherent EPPA, and stationary independence relations for the countable metrically homogeneous graphs in Cherlin’s catalogue. It introduces a canonical edge-labelled completion algorithm that preserves admissible constraints and yields EPPA-witnesses and Ramsey expansions across primitive, bipartite, and antipodal classes, with tree-like cases forming the notable exceptions. The results imply strong automorphism-group consequences for Fraïssé limits, including amenability, unique ergodicity, universal minimal flows, ample generics, small index property, Bergman property, and Serre’s FA. The paper also develops a coherent framework tying 3-constrained spaces, Henson constraints, and infinite-diameter cases, and outlines several open problems, including big Ramsey degrees and EPPA for two-graphs. Overall, it advances Nešetřil’s Ramsey-classification program by providing empirical evidence of convergent techniques across EPPA, Ramsey property, and independence relations in a wide class of metrically constrained structures.

Abstract

We investigate Ramsey expansions, the coherent extension property for partial isometries (EPPA), and the existence of a stationary independence relation for all classes of metrically homogeneous graphs from Cherlin's catalogue. We show that, with the exception of tree-like graphs, all metric spaces in the catalogue have precompact Ramsey expansions (or lifts) with the expansion property. With two exceptions we can also characterise the existence of a stationary independence relation and coherent EPPA. Our results are a contribution to Nešetřil's classification programme of Ramsey classes and can be seen as empirical evidence of the recent convergence in techniques employed to establish the Ramsey property, the expansion property, EPPA and the existence of a stationary independence relation. At the heart of our proof is a canonical way of completing edge-labelled graphs to metric spaces in Cherlin's classes. The existence of such a ``completion algorithm'' then allows us to apply several strong results in the areas that imply EPPA or the Ramsey property. The main results have numerous consequences for the automorphism groups of the Fraisse limits of the classes. As corollaries, we prove amenability, unique ergodicity, existence of universal minimal flows, ample generics, small index property, 21-Bergman property and Serre's property (FA).

Ramsey expansions of metrically homogeneous graphs

TL;DR

This work provides a near-complete characterization of Ramsey expansions, coherent EPPA, and stationary independence relations for the countable metrically homogeneous graphs in Cherlin’s catalogue. It introduces a canonical edge-labelled completion algorithm that preserves admissible constraints and yields EPPA-witnesses and Ramsey expansions across primitive, bipartite, and antipodal classes, with tree-like cases forming the notable exceptions. The results imply strong automorphism-group consequences for Fraïssé limits, including amenability, unique ergodicity, universal minimal flows, ample generics, small index property, Bergman property, and Serre’s FA. The paper also develops a coherent framework tying 3-constrained spaces, Henson constraints, and infinite-diameter cases, and outlines several open problems, including big Ramsey degrees and EPPA for two-graphs. Overall, it advances Nešetřil’s Ramsey-classification program by providing empirical evidence of convergent techniques across EPPA, Ramsey property, and independence relations in a wide class of metrically constrained structures.

Abstract

We investigate Ramsey expansions, the coherent extension property for partial isometries (EPPA), and the existence of a stationary independence relation for all classes of metrically homogeneous graphs from Cherlin's catalogue. We show that, with the exception of tree-like graphs, all metric spaces in the catalogue have precompact Ramsey expansions (or lifts) with the expansion property. With two exceptions we can also characterise the existence of a stationary independence relation and coherent EPPA. Our results are a contribution to Nešetřil's classification programme of Ramsey classes and can be seen as empirical evidence of the recent convergence in techniques employed to establish the Ramsey property, the expansion property, EPPA and the existence of a stationary independence relation. At the heart of our proof is a canonical way of completing edge-labelled graphs to metric spaces in Cherlin's classes. The existence of such a ``completion algorithm'' then allows us to apply several strong results in the areas that imply EPPA or the Ramsey property. The main results have numerous consequences for the automorphism groups of the Fraisse limits of the classes. As corollaries, we prove amenability, unique ergodicity, existence of universal minimal flows, ample generics, small index property, 21-Bergman property and Serre's property (FA).

Paper Structure

This paper contains 61 sections, 70 theorems, 45 equations, 13 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Edge-labelled graphs and metric completions
  3. Extension property for partial automorphisms (EPPA)
  4. Ramsey classes
  5. Stationary independence relations
  6. Obstacles to completion
  7. Outline of the paper
  8. Preliminaries
  9. Edge-labelled graphs as relational structures
  10. Ramsey expansions
  11. Coherent EPPA
  12. Cherlin's catalogue of metrically homogeneous graphs
  13. 3-constrained spaces $\mathcal{A}_3$
  14. Henson constraints
  15. Primitive 3-constrained spaces
  16. ...and 46 more sections

Key Result

Theorem 1.1

Let $\Gamma$ be the associated metric space of a countably infinite metrically homogeneous graph $\mathbf {G}$ of diameter $\delta$ from Cherlin's catalogue. Then one of the following applies: By "a predicate denoting the bipartition" we mean a unary predicate defining two equivalence classes of vertices, such that all distances in the same equivalence class are even. All the Ramsey expansions ab

Figures (13)

  • Figure 1:
  • Figure 2: Forks used by $\mathbb F_M$.
  • Figure 3: A sketch of the main parameters of the completion algorithm, the Optimality Lemma \ref{['lem:bestcompletion']} and the Parity Lemma \ref{['lem:sameparity']}.
  • Figure 4: Forks considered by the algorithm to complete to $\mathcal{A}^5_{3,3,16,13}$ with $M=3$.
  • Figure 5: Types of forbidden triangles.
  • ...and 8 more figures

Theorems & Definitions (168)

  • Definition 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4: (Local) Stationary Independence Relation
  • Definition 1.5
  • Definition 1.6: Triangle constraints
  • Definition 2.1: Type
  • ...and 158 more