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 introduce the relation of almost-reduction in an arbitrary topological Ramsey space R as a generalization of the relation of almost-inclusion on N [∞] . This leads us to a type of ultrafilter U ⊆ R which corresponds to the well-known notion of selective ultrafilter on N. The relationship turns out to be rather exact in the sense that it permits us to lift several well-known facts about selective ultrafilters on N and the Ellentuck space N [∞] to the ultrafilter U and the Ramsey space R. For example, we prove that the open coloring axiom holds on L(R) [U], extending therefore the result from [3] which gives the same conclusion for the Ramsey space N [∞] . PreliminariesWe follow [13] in describing what a topological Ramsey space is, rather than the earlier reference [2], where a slightly different definition is given. Consider triplets of the form (R, ≤, (r n ) n∈N ), where R is a set, ≤ is a pre-order on R, and for every n ∈ N, r n : R −→ AR n is a function with range AR n . If A ≤ B, we say that A is a reduction of B; and for each A ∈ R, we say that r n (A) is the nth approximation of A. We will assume that the following are satisfied: (A1) For any A ∈ R, r 0 (A) = ∅. (A2) For any A, B ∈ R, if A = B, then there exists n such that r n (A) = r n (B). (A3) If r n (A) = r m (B), then n = m and for all i < n, r i (A) = r i (B).These three assumptions allow us to identify each A ∈ R with the sequence (r n (A)) n of its approximations. In this way, if we consider the space AR := n AR n with the discrete topology, we can identify R with a subspace of the (metric) space AR N (with the product topology) of all the sequences of elements of AR. Via this identification, we will regard R as a subspace of AR N , and we will say that R is metrically closed if it is a closed subspace of AR N .Moreover, for a ∈ AR we define the length of a, |a|, as the unique n such that a = r n (A) for some A ∈ R. We will further identify a with the sequence {r i (A)} i≤n . So if a = r n (A) and a = r i (A) for i ≤ n, then we write a = r i (a) (that is, we extend the domain of the function r i to the set of a ∈ AR with i ≤ |a|). In this case, we also write a a and say that a is an initial segment of a. We shall also consider on R the Ellentuck type neighborhoods [a, A] = {B ∈ R : ∃n(a = r n (B)) and B ≤ A}, where a ∈ AR and A ∈ R. If [a, A] = ∅, we will say that a is compatible with A (or A is compatible with a). Let AR(A) = {a ∈ AR : a is compatible with A}. *

Given a topological Ramsey space (R,≤,r), we extend the notion of semiselective coideal to sets H⊆R and study conditions for scriptH that will enable us to make the structure (R,H,≤,r) a Ramsey space (not necessarily topological) and also study forcing notions related to scriptH which will satisfy abstract versions of interesting properties of the corresponding forcing notions in the realm of Ellentuck's space (cf. ). This extends results from to the most general context of topological Ramsey spaces. As applications, we prove that for every topological Ramsey space scriptR, under suitable large cardinal hypotheses every semiselective ultrafilter U⊆R is generic over L(R); and that given a semiselective coideal H⊆R, every definable subset of scriptR is scriptH‐Ramsey. This generalizes the corresponding results for the case when scriptR is equal to Ellentuck's space (cf. ).

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

It is shown that Matet's characterization ([9]) of the Ramsey property relative to a selective co-ideal H, in terms of games of Kastanas ([6]), still holds if we consider semiselectivity ([3]) instead of selectivity. Moreover, we prove that a co-ideal H is semiselective if and only if Matet's game-theoretic characterization of the H-Ramsey property holds. This lifts Kastanas's characterization of the classical Ramsey property to its optimal setting, from the point of view of the local Ramsey theory and gives a game-theoretic counterpart to a theorem of Farah [3], asserting that a co-ideal H is semiselective if and only if the family of H-Ramsey subsets of N [∞] coincides with the family of those sets having the abstract H-Baire property. Finally, we show that under suitable assumptions, for every semiselective co-ideal H all sets of real numbers are H-Ramsey.