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Register a new userBig Ramsey degrees and divisibility in classes of ultrametric spaces
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L. Nguyen Van The
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Given a countable set S of positive reals, we study finite-dimensional Ramsey-theoretic properties of the countable ultrametric Urysohn space with distances in S.
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We study Ramsey-theoretic properties of several natural classes of finite ultrametric spaces, describe the corresponding Urysohn spaces and compute a dynamical invariant attached to their isometry groups.
We formulate a property strengthening the Disjoint Amalgamation Property and prove that every Fraisse structure in a finite relational language with relation symbols of arity at most two having this property has finite big Ramsey degrees which have a simple characterization. It follows that any such Fraisse structure admits a big Ramsey structure. Furthermore, we prove indivisibility for every Fraisse structure in an arbitrary finite relational language satisfying this property. This work offers a streamlined and unifying approach to Ramsey theory on some seemingly disparate classes of Fraisse structures. Novelties include a new formulation of coding trees in terms of 1-types over initial segments of the Fraisse structure, and a direct characterization of the degrees without appeal to the standard method of envelopes.
Building on previous work of the author, for each finite triangle-free graph $mathbf{G}$, we determine the equivalence relation on the copies of $mathbf{G}$ inside the universal homogeneous triangle-free graph, $mathcal{H}_3$, with the smallest number of equivalence classes so that each one of the classes persists in every isomorphic subcopy of $mathcal{H}_3$. This characterizes the exact big Ramsey degrees of $mathcal{H}_3$. It follows that the triangle-free Henson graph is a big Ramsey structure.
We build a collection of topological Ramsey spaces of trees giving rise to universal inverse limit structures, extending Zhengs work for the profinite graph to the setting of Fra{i}ss{e} classes of finite ordered binary relational structures with the Ramsey property. This work is based on the Halpern-L{a}uchli theorem, but different from the Milliken space of strong subtrees. Based on these topological Ramsey spaces and the work of Huber-Geschke-Kojman on inverse limits of finite ordered graphs, we prove that for each such Fra{i}ss{e} class, its universal inverse limit structures has finite big Ramsey degrees under finite Baire-measurable colourings. For finite ordered graphs, finite ordered $k$-clique free graphs ($kgeq 3$), finite ordered oriented graphs, and finite ordered tournaments, we characterize the exact big Ramsey degrees.
As a result of 33 intercontinental Zoom calls, we characterise big Ramsey degrees of the generic partial order in a similar way as Devlin characterised big Ramsey degrees of the generic linear order (the order of rationals).
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