Topological Ramsey spaces from Fraïssé classes, Ramsey-classification theorems, and initial structures in the Tukey types of p-points

Abstract

A general method for constructing a new class of topological Ramsey spaces is presented. Members of such spaces are infinite sequences of products of Fraïssé classes of finite relational structures satisfying the Ramsey property. The Product Ramsey Theorem of Sokič is extended to equivalence relations for finite products of structures from Fraïssé 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 Pudlák-Rödl 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 Fraïssé 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 Fraïssé 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 $[ω]^{<ω}$. In contrast, the set of isomorphism types of any product of finitely many Fraïssé 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.