64,174 research outputs found

    Topological Ramsey spaces from Fra\"iss\'e classes, Ramsey-classification theorems, and initial structures in the Tukey types of p-points

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    A general method for constructing a new class of topological Ramsey spaces is presented. Members of such spaces are infinite sequences of products of Fra\"iss\'e classes of finite relational structures satisfying the Ramsey property. The Product Ramsey Theorem of Soki\v{c} is extended to equivalence relations for finite products of structures from Fra\"iss\'e classes of finite relational structures satisfying the Ramsey property and the Order-Prescribed Free Amalgamation Property. This is essential to proving Ramsey-classification theorems for equivalence relations on fronts, generalizing the Pudl\'ak-R\"odl Theorem to this class of topological Ramsey spaces. To each topological Ramsey space in this framework corresponds an associated ultrafilter satisfying some weak partition property. By using the correct Fra\"iss\'e classes, we construct topological Ramsey spaces which are dense in the partial orders of Baumgartner and Taylor in \cite{Baumgartner/Taylor78} generating p-points which are kk-arrow but not k+1k+1-arrow, and in a partial order of Blass in \cite{Blass73} producing a diamond shape in the Rudin-Keisler structure of p-points. Any space in our framework in which blocks are products of nn many structures produces ultrafilters with initial Tukey structure exactly the Boolean algebra P(n)\mathcal{P}(n). If the number of Fra\"iss\'e classes on each block grows without bound, then the Tukey types of the p-points below the space's associated ultrafilter have the structure exactly [ω]<ω[\omega]^{<\omega}. In contrast, the set of isomorphism types of any product of finitely many Fra\"iss\'e classes of finite relational structures satisfying the Ramsey property and the OPFAP, partially ordered by embedding, is realized as the initial Rudin-Keisler structure of some p-point generated by a space constructed from our template.Comment: 35 pages. Abstract and introduction re-written to make very clear the main points of the paper. Some typos and a few minor errors have been fixe

    Independence densities of hypergraphs

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    We consider the number of independent sets in hypergraphs, which allows us to define the independence density of countable hypergraphs. Hypergraph independence densities include a broad family of densities over graphs and relational structures, such as FF-free densities of graphs for a given graph F.F. In the case of kk-uniform hypergraphs, we prove that the independence density is always rational. In the case of finite but unbounded hyperedges, we show that the independence density can be any real number in [0,1].[0,1]. Finally, we extend the notion of independence density via independence polynomials

    The Quantum Monad on Relational Structures

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    Homomorphisms between relational structures play a central role in finite model theory, constraint satisfaction, and database theory. A central theme in quantum computation is to show how quantum resources can be used to gain advantage in information processing tasks. In particular, non-local games have been used to exhibit quantum advantage in boolean constraint satisfaction, and to obtain quantum versions of graph invariants such as the chromatic number. We show how quantum strategies for homomorphism games between relational structures can be viewed as Kleisli morphisms for a quantum monad on the (classical) category of relational structures and homomorphisms. We use these results to exhibit a wide range of examples of contextuality-powered quantum advantage, and to unify several apparently diverse strands of previous work
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