A notion of selective ultrafilter corresponding to topological Ramsey spaces

Abstract

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}. *

Local Ramsey theory: an abstract approach

“…). This extends results from to the most general context of topological Ramsey spaces. As applications, we prove that for every topological Ramsey space , under suitable large cardinal hypotheses every semiselective ultrafilter is generic over ; and that given a semiselective coideal , every definable subset of is ‐Ramsey.…”

“…Farah showed not only that the semiselectivity of is enough to make the ‐Ramsey property equivalent to the abstract Baire property with respect to , but also showed that this latter equivalence characterizes semiselectivity. In , a step toward the understanding of the local Ramsey property within the most general context of topological Ramsey spaces was taken.…”

“…In this work, given a topological Ramsey space , we extend the notion of semiselective coideal to sets and study conditions for that will enable us to make the structure a Ramsey space (not necessarily topological) and also study forcing notions related to which will satisfy abstract versions of interesting properties of the corresponding forcing notions in the realm of Ellentuck's space, extending results from to the more general context of topological Ramsey spaces. As applications, we prove that for every topological Ramsey space , under suitable large cardinal hypotheses every semiselective ultrafilter is generic over ; and that given a semiselective coideal , every definable subset of is ‐Ramsey.…”

On Galvin’s Lemma and Ramsey Spaces

“…Using Theorem 3.1, we give a simpler proof of [13,Theorem 1.7], which is an abstract version of Ramsey's theorem.…”

“…In the proof of Theorem 3.1, a technique of selection by diagonalization (or by fusion) is used recurrently; see for example, the proof of Claim 3.3. We can now attempt to isolate from it a notion of abstract selective coideal analog to the concept of selective coideal on N (see [11]) to generalize the results contained in [13], where a notion selective ultrafilter corresponding to topological Ramsey spaces is given. This in turn could lead us to an abstract approach to local Ramsey theory.…”