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Structurable equivalence relations
"... (A) A countable Borel equivalence relation on a standard Borel space X is a Borel equivalence relation E ⊆ X2 with the property that every equivalence class [x]E, x ∈ X, is countable. Over the last 25 years there has been an extensive study of countable Borel equivalence relations and their connecti ..."
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(A) A countable Borel equivalence relation on a standard Borel space X is a Borel equivalence relation E ⊆ X2 with the property that every equivalence class [x]E, x ∈ X, is countable. Over the last 25 years there has been an extensive study of countable Borel equivalence relations and their connection
Invariant measures via inverse limits of finite structures, ArXiv eprint 1310.8147
, 2013
"... Abstract. Building on recent results regarding symmetric probabilistic constructions of countable structures, we provide a method for constructing probability measures, concentrated on certain classes of countably infinite structures, that are invariant under all permutations of the underlying se ..."
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Abstract. Building on recent results regarding symmetric probabilistic constructions of countable structures, we provide a method for constructing probability measures, concentrated on certain classes of countably infinite structures, that are invariant under all permutations of the underlying set that fix all constants. These measures are constructed from inverse limits of measures on certain finite structures. We use this construction to obtain invariant probability measures concentrated on the classes of countable models of certain firstorder theories, including measures that do not assign positive measure to the isomorphism class of any single model. We also characterize those transitive Borel Gspaces admitting a Ginvariant probability measure, when G is an arbitrary countable product of symmetric groups on a countable set.
A CLASSIFICATION OF ORBITS ADMITTING A UNIQUE INVARIANT MEASURE
"... Abstract. The group S ∞ acts via the logic action on the space of countable structures in a given countable language that have a fixed underlying set. We consider the number of ergodic probability measures on this space that are invariant under the logic action and are concentrated on the isomorph ..."
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Abstract. The group S ∞ acts via the logic action on the space of countable structures in a given countable language that have a fixed underlying set. We consider the number of ergodic probability measures on this space that are invariant under the logic action and are concentrated on the isomorphism class of a particular structure. We show that this number must be either zero, or one, or continuum. Further, such an isomorphism class admits a unique S∞invariant probability measure precisely when the structure is highly homogeneous; by a result of Peter J. Cameron, these are the structures that are interdefinable with one of the five reducts of the rational
Bowtiefree graphs have a Ramsey lift
, 2013
"... A bowtie is a graph consisting of two triangles with one vertex identified. We show that the class of all (countable) graphs not containing a bowtie as a subgraph have a Ramsey lift (expansion). This is the first nontrivial Ramsey class with a nontrivial algebraic closure ..."
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A bowtie is a graph consisting of two triangles with one vertex identified. We show that the class of all (countable) graphs not containing a bowtie as a subgraph have a Ramsey lift (expansion). This is the first nontrivial Ramsey class with a nontrivial algebraic closure
Bowtiefree graphs have a Ramsey lift
"... A bowtie is a graph consisting of two triangles with one vertex identified. We show that the class of all (countable) graphs not containing a bowtie as a subgraph have a Ramsey lift (expansion). This is the first nontrivial Ramsey class with a nontrivial algebraic closure. 1 ..."
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A bowtie is a graph consisting of two triangles with one vertex identified. We show that the class of all (countable) graphs not containing a bowtie as a subgraph have a Ramsey lift (expansion). This is the first nontrivial Ramsey class with a nontrivial algebraic closure. 1