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 set\v ril'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).

翻译：我们研究Cherlin分类目录中所有度量齐次图类的Ramsey展开、部分等距的相干扩张性质（EPPA）以及平稳独立关系的存在性。我们证明，除树状图外，目录中所有度量空间均存在具有扩张性质的预紧Ramsey展开（或提升）。除两个例外情形外，我们还能刻画平稳独立关系与相干EPPA的存在性。这些结果是对Nešetřil的Ramsey类分类纲领的贡献，并可视为近期在建立Ramsey性质、扩张性质、EPPA及平稳独立关系存在性方面技术趋同性的经验证据。证明的核心在于将边标号图规范地完备化为Cherlin类中的度量空间。这种"完备化算法"的存在性使我们能够应用该领域中若干强结果，从而推导出EPPA或Ramsey性质。主要结果对各类Fraïssé极限的自同构群产生多重影响。作为推论，我们证明了顺从性、唯一遍历性、通用极小流的存在性、充足泛型性、小指标性质、21-伯格曼性质以及Serre性质(FA)。