We discuss the Ramsey property, the existence of a stationary independence relation and the coherent extension property for partial isometries (coherent EPPA) for all classes of metrically homogeneous graphs from Cherlin's catalogue, which is conjectured to include all such structures. 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 the coherent EPPA.Our results can be seen as a new contribution to Nešetřil's classification programme of Ramsey classes and as empirical evidence of the recent convergence in techniques employed to establish the Ramsey property, the expansion (or lift or ordering) 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 and respectively the Ramsey property.The main results have numerous corollaries on the automorphism groups of the Fraïssé limits of the classes, such as amenability, unique ergodicity, existence of universal minimal flows, ample generics, small index property, 21-Bergman property and Serre's property (FA).

We prove that certain classes of metrically homogeneous graphs omitting triangles of odd short perimeter as well as triangles of long perimeter have the extension property for partial automorphisms and we describe their Ramsey expansions.Dedicated to Norbert Sauer on the occasion of his 70th birthday.

We show that the independence number of a countably infinite connected HHhomogeneous graph that does not contain the Rado graph as a spanning subgraph is finite and present a classification of MB-homogeneous graphs up to bimorphism-equivalence as a consequence.

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Our goal is to investigate a close relative of the independent transversal problem in the class of infinite Kn-free graphs: we show that for any infinite Kn-free graph G " pV, Eq and m P N there is a minimal r " rpG, mq such that for any balanced r-colouring of the vertices of G one can find an independent set which meets at least m colour classes in a set of size |V |. Answering a conjecture of S. Thomassé, we express the exact value of rpHn, mq (using Ramsey-numbers for finite digraphs), where Hn is Henson's countable universal homogeneous Kn-free graph. In turn, we deduce a new partition property of Hn regarding balanced embeddings of bipartite graphs: for any finite bipartite G with bipartition A, B, if the vertices of Hn are partitioned into two infinite classes then there is an induced copy of G in Hn such that the images of A and B are contained in different classes.