Parametrizing the abstract Ellentuck theorem

We give a parametrization with perfect sets of the abstract Ellentuck theorem. The main tool for achieving this goal is a sort of parametrization of an abstract version of the Nash-Williams theorem. As corollaries, we obtain some known classical results like the parametrized version of the Galvin-Prikry theorem due to Miller and Todorcevic, and the parametrized version of Ellentuck's theorem due to Pawlikowski. Also, we obtain parametized vesions of nonclassical results such as Milliken's theorem.Comment: 16 pages. Submitted to Elsevier's Discrete Mathematic

On Galvin's lemma and Ramsey spaces

An abstract version of Galvin's lemma is proven, within the framework of the theory of Ramsey spaces. Some instances of it are explored.Comment: Accepted in Annals of Combinatoric

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

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 $k$-arrow but not $k+1$-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 $n$ many structures produces ultrafilters with initial Tukey structure exactly the Boolean algebra $\mathcal{P}(n)$. 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 $[\omega]^{<\omega}$. 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