20 research outputs found
Injective objects and retracts of Fra\"iss\'e limits
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
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
We introduce a general method for showing under weak forcing axioms that
reduced products of countable models of a theory 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
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