Topological Ramsey Spaces Dense in Forcings

Abstract

Topological Ramsey spaces are spaces which support infinite dimensional Ramsey theory similarly to the Ellentuck space. Each topological Ramsey space is endowed with a partial ordering which can be modified to a σ-closed 'almost reduction' relation analogously to the partial ordering of 'mod finite' on [ω] ω . Such forcings add new ultrafilters satisfying weak partition relations and have complete combinatorics. In cases where a forcing turned out to be equivalent to a topological Ramsey space, the strong Ramsey-theoretic techniques have aided in a fine-tuned analysis of the Rudin-Keisler and Tukey structures associated with the forced ultrafilter and in discovering new ultrafilters with complete combinatorics. This expository paper provides an overview of this collection of results and an entry point for those interested in using topological Ramsey space techniques to gain finer insight into ultrafilters satisfying weak partition relations.

Ramsey degrees of ultrafilters, pseudointersection numbers, and the tools of topological Ramsey spaces

“…The initial purpose for constructing those new Ramsey spaces was to find the exact Rudin-Keisler and Tukey structures below those ultrafilters. The Abstract Ellentuck Theorem proved to be vital to those investigations, which have been the subject of work in [20], [21], [18], [14], and [15]; the paper [16] provides an overview those results.…”

“…Although Blass had already shown such ultrafilters exist (see [6]), the point of P 1 was to construct a weakly Ramsey ultrafilter with complete combinatorics, analagous to the result that any Ramsey ultrafilter in the model V [G] obtained by Lévy collapsing a Mahlo cardinal to ℵ 1 is ([ω] ω , ⊆ * )-generic over HOD(R) V [G] (see [8] and [33]). One of the advantages of forcing with topological Ramsey spaces is that the associated ultrafilter automatically has complete combinatorics in the presence of large cardinals (see [13] for the result and [16] for an overview of this area). In [20], a topological Ramsey space denoted R 1 was constructed which forms a dense subset of Laflamme's forcing P 1 , hence generating the same weakly Ramsey ultrafilter.…”

“…Such Ramsey spaces were used to find exact Rudin-Keisler and Tukey structures below those ultrafilters. An overview of this area can be found in [16].…”