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

We present a general method for constructing a new class of topological Ramsey spaces. Members of such spaces are infinite sequences of products of Fräıssé classes of finite relational structures satisfying the Ramsey property. We extend the Product Ramsey Theorem of Sokič to equivalence relations for finite products of structures from Fräıssé classes of finite relational structures satisfying the Ramsey property. This is then applied to prove Ramsey-classification theorems for equivalence relations on fronts, generalizing the Pudlák-Rödl Theorem to our class of topological Ramsey spaces. To each topological Ramsey space in this framework correspond associated ultrafilters satisfying weak partition properties. The Ramsey-classification theorems are applied to classify the structure of the Tukey types of all ultrafilters Tukey reducible to the associated ultrafilter. Furthermore, the structure of the Rudin-Keisler classes inside each Tukey type are also completely classified in terms of the embedding relation on the Fräıssé classes. Examples of topological Ramsey spaces constructed from our method include the following. We construct spaces which generate p-points which are k-arrow but not k + 1-arrow, simplifying constructions of Baumgartner and Taylor in [2]. We also construct hypercube spaces Hn, 2 ≤ n < ω, generalizing a construction of Blass in [3]. Each Hn, and more generally any space in our framework in which blocks are products of n many structures, produces ultrafilters with Tukey structures below them which are exactly the n-element Boolean algebra P(n). These are the first examples of initial structures in the Tukey types of ultrafilters which are not simply linear chains. Furthermore, we construct a space Hω for which the Tukey types of the p-points below its associated ultrafilter have the structure exactly [ω]<ω .