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 [2] generating p-points which are k-arrow but not k + 1-arrow, and in a partial order of Blass in [3] 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 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.
We give a partial answer to the following question of Dobrinen: For a given topological Ramsey space R, are the notions of selective for R and Ramsey for R equivalent? Every topological Ramsey space R has an associated notion of Ramsey ultrafilter for R and selective ultrafilter for R (see [1]). If R is taken to be the Ellentuck space then the two concepts reduce to the familiar notions of Ramsey and selective ultrafilters on ω; so by a wellknown result of Kunen the two are equivalent. We give the first example of an ultrafilter on a topological Ramsey space that is selective but not Ramsey for the space, and in fact a countable collection of such examples.For each positive integer n we show that for the topological Ramsey space R n from [2], the notions of selective for R n and Ramsey for R n are not equivalent. In particular, we prove that forcing with a closely related space using almost-reduction, adjoins an ultrafilter that is selective but not Ramsey for R n . Moreover, we introduce a notion of finite product among members of the family {R n : n < ω}. We show that forcing with closely related product spaces using almost-reduction, adjoins ultrafilters that are selective but not Ramsey for these product topological Ramsey spaces.
Associated to each ultrafilter U on ω and each map p : ω → ω is a Dedekind cut in the ultrapower ω ω / p(U). Blass has characterized, under CH, the cuts obtainable when U is taken to be either a p-point ultrafilter, a weakly-Ramsey ultrafilter or a Ramsey ultrafilter. Dobrinen and Todorčević have introduced the topological Ramsey space R 1 . Associated to the space R 1 is a notion of Ramsey ultrafilter for R 1 generalizing the familiar notion of Ramsey ultrafilter on ω. We characterize, under CH, the cuts obtainable when U is taken to be a Ramsey for R 1 ultrafilter and p is taken to be any map. In particular, we show that the only cut obtainable is the standard cut, whose lower half consists of the collection of equivalence classes of constants maps. Forcing with R 1 using almost-reduction adjoins an ultrafilter which is Ramsey for R 1 . For such ultrafilters U 1 , Dobrinen and Todorčević have shown that the Rudin-Keisler types of the p-points within the Tukey type of U 1 consists of a strictly increasing chain of rapid p-points of order type ω. We show that for any Rudin-Keisler mapping between any two p-points within the Tukey type of U 1 the only cut obtainable is the standard cut. These results imply existence theorems for special kinds of ultrafilters.
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