Ramsey expansions of metrically homogeneous graphs

TL;DR

This paper studies Ramsey expansions, EPPA, and stationary independence in metrically homogeneous graphs, providing a comprehensive classification and new completion algorithms that impact automorphism group properties.

Contribution

It offers a classification of Ramsey expansions and related properties for all classes of metrically homogeneous graphs, introducing a canonical completion algorithm.

Findings

Most metric spaces in Cherlin's catalogue have precompact Ramsey expansions.

Characterization of stationary independence relation and EPPA for these classes.

Implications for automorphism groups include amenability, ergodicity, and property (FA).

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 Ne\v{s}et\v{r}il'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).