Ramsey expansions of metrically homogeneous graphs

Abstract

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).

Big Ramsey degrees and forbidden cycles

“…Vertex partition properties of Urysohn spaces were extensively studied in connection to oscillation stability [10] and determining their big Ramsey degrees presented a long standing open problem: Corollary 1 (i) is a special case of the main result of [12], (ii) is a strengthening of [7, Corollary 6.5 (3)], (iii) strengthens [9] and (iv) is a strengthening of [1] to infinite structures.…”

“…Possible obstacles to completing a structure in this language to a metric space are irreducible substructures with at most 2 vertices and induced non-metric cycles. These are cycles with the longest edge of a length exceeding the sum of the lengths of all the remaining edges; see [1].…”

Ramsey properties and extending partial automorphisms for classes of finite structures

“…The construction above adds an extra tool to the existing constructions of EPPA-extensions and can be thus used as an additional layer in the construction of EPPA-extensions for non-free amalgamation classes based on application of Herwig-Lascar theorem [14,38,33]. An example of such application is given in [3] giving EPPA for some classes of antipodal metric spaces.…”

“…On the structural Ramsey theory side open problems include Ramsey properties of finite lattices and other algebraic structures where the axioms (such as associativity) are difficult to control in an amalgamation procedure. See [20,3] for results on Ramsey classes.…”

Extending partial automorphisms of $n$-partite tournaments

“…After this, the quest of identifying new classes of structures with EPPA continued. In 2000, Herwig and Lascar [5] combined advanced techniques from combinatorics, model theory and group theory and gave a structural condition which was later used to identify many additional examples of classes having EPPA (such as various classes of metric spaces [17,10,1,13]).…”