20 research outputs found

    Injective objects and retracts of Fra\"iss\'e limits

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    We present a purely category-theoretic characterization of retracts of Fra\"iss\'e limits. For this aim, we consider a natural version of injectivity with respect to a pair of categories (a category and its subcategory). It turns out that retracts of Fra\"iss\'e limits are precisely the objects that are injective relatively to such a pair. One of the applications is a characterization of non-expansive retracts of Urysohn's universal metric space.Comment: 33 pages; minor corrections; extended introduction and more references. Final versio

    Constraint satisfaction problems for reducts of homogeneous graphs

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    For n >= 3, let (H-n, E) denote the nth Henson graph, i.e., the unique countable homogeneous graph with exactly those finite graphs as induced subgraphs that do not embed the complete graph on n vertices. We show that for all structures Gamma with domain H-n whose relations are first-order definable in (H-n, E) the constraint satisfaction problem for F either is in P or is NP-complete. We moreover show a similar complexity dichotomy for all structures whose relations are first-order definable in a homogeneous graph whose reflexive closure is an equivalence relation. Together with earlier results, in particular for the random graph, this completes the complexity classification of constraint satisfaction problems of structures first-order definable in countably infinite homogeneous graphs: all such problems are either in P or NP-complete

    Trivial Automorphisms of Reduced Products

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    We introduce a general method for showing under weak forcing axioms that reduced products of countable models of a theory TT have as few automorphisms as possible. We show that such forcing axioms imply that reduced products of countably infinite or finite fields, linear orders, trees, or random graphs have only trivial automorphisms.Comment: 36 page

    Multicoloured Random Graphs: Constructions and Symmetry

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    This is a research monograph on constructions of and group actions on countable homogeneous graphs, concentrating particularly on the simple random graph and its edge-coloured variants. We study various aspects of the graphs, but the emphasis is on understanding those groups that are supported by these graphs together with links with other structures such as lattices, topologies and filters, rings and algebras, metric spaces, sets and models, Moufang loops and monoids. The large amount of background material included serves as an introduction to the theories that are used to produce the new results. The large number of references should help in making this a resource for anyone interested in beginning research in this or allied fields.Comment: Index added in v2. This is the first of 3 documents; the other 2 will appear in physic
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