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Forbidden cycles in metrically homogeneous graphs

Jan Hubička, Michael Kompatscher, Matěj Konečný

TL;DR

This work delivers an explicit description of the forbidden cycle set for primitive 3-constrained metrically homogeneous graphs of finite diameter by extending the prior finite-cycle framework. Building on the magic completion algorithm, the authors define four natural families of cycles plus non-metric cycles, and prove that the metric-edge-labelled graphs form the class $\mathcal{G}^\delta_{K_1,K_2,C_0,C_1} = \mathrm{Forb}(\mathcal{F})$ with $\mathcal{F}$ closed under algorithmic steps and inverse steps. The result unifies and completes previous Ramsey-expansion and EPPA-related analyses, and enables reinterpretations in terms of semigroup-valued metrics and ω-categoricity through $(1,\delta)$-graph homogenizations. The explicit cycle descriptions illuminate the structure of Cherlin’s catalogue in the 3-constrained primitive case and provide concrete tools for further applications in structural Ramsey theory and topological dynamics.

Abstract

In a recent paper by a superset of the authors it was proved that for every primitive 3-constrained space $Γ$ of finite diameter $δ$ from Cherlin's catalogue of metrically homogeneous graphs, there exists a finite family $\mathcal F$ of $\{1,\ldots, δ\}$-edge-labelled cycles such that a $\{1,\ldots, δ\}$-edge-labelled graph is a subgraph of $Γ$ if and only if it contains no homomorphic images of cycles from $\mathcal F$. However, the cycles in the families $\mathcal F$ were not described explicitly as it was not necessary for the analysis of Ramsey expansions and the extension property for partial automorphisms. This paper fills this gap by providing an explicit description of the cycles in the families $\mathcal F$, heavily using the previous result in the process. Additionally, we explore the potential applications of this result, such as interpreting the graphs as semigroup-valued metric spaces or homogenizations of $ω$-categorical $\{1,δ\}$-edge-labelled graphs.

Forbidden cycles in metrically homogeneous graphs

TL;DR

This work delivers an explicit description of the forbidden cycle set for primitive 3-constrained metrically homogeneous graphs of finite diameter by extending the prior finite-cycle framework. Building on the magic completion algorithm, the authors define four natural families of cycles plus non-metric cycles, and prove that the metric-edge-labelled graphs form the class with closed under algorithmic steps and inverse steps. The result unifies and completes previous Ramsey-expansion and EPPA-related analyses, and enables reinterpretations in terms of semigroup-valued metrics and ω-categoricity through -graph homogenizations. The explicit cycle descriptions illuminate the structure of Cherlin’s catalogue in the 3-constrained primitive case and provide concrete tools for further applications in structural Ramsey theory and topological dynamics.

Abstract

In a recent paper by a superset of the authors it was proved that for every primitive 3-constrained space of finite diameter from Cherlin's catalogue of metrically homogeneous graphs, there exists a finite family of -edge-labelled cycles such that a -edge-labelled graph is a subgraph of if and only if it contains no homomorphic images of cycles from . However, the cycles in the families were not described explicitly as it was not necessary for the analysis of Ramsey expansions and the extension property for partial automorphisms. This paper fills this gap by providing an explicit description of the cycles in the families , heavily using the previous result in the process. Additionally, we explore the potential applications of this result, such as interpreting the graphs as semigroup-valued metric spaces or homogenizations of -categorical -edge-labelled graphs.

Paper Structure

This paper contains 14 sections, 34 theorems, 72 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Motivation
  3. The primitive 3-constrained metrically homogeneous graphs
  4. Our results
  5. The magic completion algorithm
  6. Proof strategy
  7. $K_1$-cycles
  8. Metric $K_2$-cycles with parameters from Case \ref{['III']}
  9. Non-$C$ $K_2$-cycles with parameters from Case \ref{['II']}
  10. Non-metric cycles
  11. $C_0$- and $C_1$- cycles with $n = 1$, Case \ref{['III']} and $C'>C+1$
  12. $C_0$-cycles and $C_1$-cycles with $n = 1$ or the cycle $(5,5,5,5,5)$, Case \ref{['IIb']}
  13. $C$-cycles with $n\geq 1$ when $C'=C+1$
  14. ($1,\delta$)-graphs

Key Result

Theorem 1.3

Let $(\delta, K_1, K_2, C_0, C_1)$ be an acceptable sequence of parameters (in particular, $\delta\geq 3$). Then the associated class $\mathcal{A}^\delta_{K_1,K_2,C_0,C_1}$ is an amalgamation class if and only if one of the following two groups of conditions is satisfied, where we write $C$ for $\mi

Theorems & Definitions (76)

  • Definition 1.1: Triangle constraints
  • Definition 1.2: Acceptable numerical parameters
  • Theorem 1.3: Cherlin's Admissibility Theorem Cherlin2013 (simplified)
  • Definition 1.4: Forbidden cycles
  • Theorem 1.5
  • Definition 2.1: Magic distances
  • Definition 2.3: Magic semigroup
  • Definition 2.5: Time function and the magic permutation
  • Definition 2.6: The magic completion algorithm
  • Theorem 2.7: Aranda2017
  • ...and 66 more