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).

翻译：我们研究了谢林目录中所有度量齐次图类的拉姆西扩张、部分等距的相干扩展性质（EPPA）以及平稳独立关系的存在性。研究表明，除树状图外，目录中所有度量空间均具有带扩张性质的预紧拉姆西扩张（或提升）。除两个例外情况外，我们还能刻画平稳独立关系与相干EPPA的存在性。这些成果是对内谢特里尔拉姆西类分类计划的贡献，并可作为近期确立拉姆西性、扩张性质、EPPA及平稳独立关系存在性的技术趋同的经验证据。证明的核心在于将边标记图典范完备为谢林类中度量空间的方法。此类“完备算法”的存在性使我们能应用多个强结果领域中的定理来推导EPPA或拉姆西性。主要结论对各类弗拉塞极限的自同构群具有多重推论，作为推论我们证明了阿梅纳性、唯一遍历性、通用极小流存在性、充裕生成元性、小指数性质、21-伯格曼性质及塞尔性质（FA）。