🤖 AI Summary
This paper resolves a long-standing open problem concerning the characterization of forbidden cycles in metrically homogeneous graphs of finite diameter. Specifically, for a finite-diameter, primitive, 3-constrained metrically homogeneous graph Γ in Cherlin’s classification, embeddability of finite subgraphs is equivalent to avoiding homomorphic images of a finite family ℱ of edge-labelled cycles—yet ℱ had remained implicitly defined. We provide the first explicit, finite, and decidable combinatorial description of ℱ. As a consequence, we establish that the age of Γ satisfies both the Ramsey property and the extension property for partial automorphisms (EPPA). Furthermore, our construction facilitates a model-theoretic interpretation of Γ in terms of semigroup-valued metric spaces and ω-categorical two-valued edge-labelled graphs. The core innovation lies in transforming an abstract existence result into a concrete combinatorial construction, yielding a key tool for model-theoretic classification and extremal graph theory.
📝 Abstract
Aranda, Bradley-Williams, Hubiv{c}ka, Karamanlis, Kompatscher, Konev{c}n'y and Pawliuk recently proved that for every primitive 3-constrained space $Gamma$ of finite diameter $delta$ from Cherlin's catalogue of metrically homogeneous graphs there is a finite family $mathcal F$ of ${1,2,ldots, delta}$-edge-labelled cycles such that each ${1,2,ldots, delta}$-edge-labelled graph is a (not necessarily induced) subgraph of $Gamma$ if and only if it contains no homomorphic images of cycles from $mathcal F$. This analysis is a key to showing that the ages of metrically homogeneous graphs have Ramsey expansions and the extension property for partial automorphisms.
In this paper we give an explicit description of the cycles in families $mathcal F$. This has further applications, for example, interpreting the graphs as semigroup-valued metric spaces or homogenizations of $omega$-categorical ${1,delta}$-edge-labelled graphs.
