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Preprint Number 1251
1251. Andrés Aranda, David Bradley-Williams, Jan Hubička, Miltiadis Karamanlis, Michael Kompatscher, Matěj Konečný, Micheal Pawliuk
Ramsey expansions of metrically homogeneous graphs
Submission date: 9 July 2017
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).
Mathematics Subject Classification: 05D10, 20B27, 54E35, Secondary: 03C15, 22F50, 37B05 ACM-class: G.2.2, F.4.1
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