Topological Ramsey Spaces Research Articles

Big Ramsey degrees in universal inverse limit structures

We build a collection of topological Ramsey spaces of trees giving rise to universal inverse limit structures, extending Zheng’s work for the profinite graph to the setting of Fraïssé classes of finite ordered binary relational structures with the Ramsey property. This work is based on the Halpern-Läuchli theorem, but different from the Milliken space of strong subtrees. Based on these topological Ramsey spaces and the work of Huber-Geschke-Kojman on inverse limits of finite ordered graphs, we prove that for each such Fraïssé class, its universal inverse limit structure has finite big Ramsey degrees under finite Baire-measurable colorings. For such Fraïssé classes satisfying free amalgamation as well as finite ordered tournaments and finite partial orders with a linear extension, we characterize the exact big Ramsey degrees.

The space of infinite partitions of $\mathbb{N}$ as a topological Ramsey space

The space of infinite partitions of $\mathbb{N}$ as a topological Ramsey space

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

This paper investigates properties of \(\sigma \)-closed forcings which generate ultrafilters satisfying weak partition relations. The Ramsey degree of an ultrafilter \({\mathcal {U}}\) for n-tuples, denoted \(t({\mathcal {U}},n)\), is the smallest number t such that given any \(l\ge 2\) and coloring \(c:[\omega ]^n\rightarrow l\), there is a member \(X\in {\mathcal {U}}\) such that the restriction of c to \([X]^n\) has no more than t colors. Many well-known \(\sigma \)-closed forcings are known to generate ultrafilters with finite Ramsey degrees, but finding the precise degrees can sometimes prove elusive or quite involved, at best. In this paper, we utilize methods of topological Ramsey spaces to calculate Ramsey degrees of several classes of ultrafilters generated by \(\sigma \)-closed forcings. These include a hierarchy of forcings due to Laflamme which generate weakly Ramsey and weaker rapid p-points, forcings of Baumgartner and Taylor and of Blass and generalizations, and the collection of non-p-points generated by the forcings \({\mathcal {P}}(\omega ^k)/\mathrm {Fin}^{\otimes k}\). We provide a general approach to calculating the Ramsey degrees of these ultrafilters, obtaining new results as well as streamlined proofs of previously known results. In the second half of the paper, we calculate pseudointersection and tower numbers for these \(\sigma \)-closed forcings and their relationships with the classical pseudointersection number \({\mathfrak {p}}\).

Topological Ramsey spaces of equivalence relations and a dual Ramsey theorem for countable ordinals

We define a collection of topological Ramsey spaces consisting of equivalence relations on ω with the property that the minimal representatives of the equivalence classes alternate according to a fixed partition of ω. To prove the associated pigeonhole principles, we make use of the left-variable Hales-Jewett theorem and its extension to an infinite alphabet. We also show how to transfer the corresponding infinite-dimensional Ramsey results to equivalence relations on countable limit ordinals (up to a necessary restriction on the set of minimal representatives of the equivalence classes) in order to obtain a dual Ramsey theorem for such ordinals.

A collection of topological Ramsey spaces of trees and their application to profinite graph theory

We construct a collection of new topological Ramsey spaces of trees. It is based on the Halpern-Lauchli theorem, but different from the Milliken space of strong subtrees. We give an example of its application by proving a partition theorem for profinite graphs.

Preserved under Sacks forcing again?

A hierarchy of topological Ramsey spaces \({\mathcal{R}_\alpha}\) (\({\alpha < \omega_1}\)), generalizing the Ellentuck space, were built by Dobrinen and Todorcevic in order to completely classify certain equivalent classes of ultrafilters Tukey (resp. Rudin–Keisler) below \({\mathcal{U}_\alpha}\) \({(\alpha < \omega_1)}\), where \({\mathcal{U}_\alpha}\) are ultrafilters constructed by Laflamme satisfying certain partition properties and have complete combinatorics over the Solovay model. We show that Nash–Williams, or Ramsey ultrafilters in these spaces are preserved under countable-support side-by-side Sacks forcing. This is achieved by proving a parametrized theorem for these spaces, and showing that Nash–Williams ultrafilters localizes the theorem. We also show that every Nash–Williams ultrafilter in \({\mathcal{R}_\alpha}\) is selective.

Local Ramsey theory: an abstract approach

AbstractGiven 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 (cf. [][]). 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. This generalizes the corresponding results for the case when is equal to Ellentuck's space (cf. [][]).

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

A general method for constructing a new class of topological Ramsey spaces is presented. Members of such spaces are infinite sequences of products of Fraisse classes of finite relational structures satisfying the Ramsey property. The Product Ramsey Theorem of Sokic is extended to equivalence relations for finite products of structures from Fraisse 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 Pudlak–Rodl 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 Fraisse classes, we construct topological Ramsey spaces which are dense in the partial orders of Baumgartner and Taylor (Trans Am Math Soc 241:283–309, 1978) generating p-points which are k-arrow but not $$k+1$$ -arrow, and in a partial order of Blass (Trans Am Math Soc 179:145–166, 1973) 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 $$\mathcal {P}(n)$$ . If the number of Fraisse 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 $$[\omega ]^{<\omega }$$ . In contrast, the set of isomorphism types of any product of finitely many Fraisse 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.

Creature forcing and topological Ramsey spaces

This article introduces a line of investigation into connections between creature forcings and topological Ramsey spaces. Three examples of sets of pure candidates for creature forcings are shown to contain dense subsets which are actually topological Ramsey spaces. A new variant of the product tree Ramsey theorem is proved in order to obtain the pigeonhole principles for two of these examples.

Infinite-dimensional Ellentuck spaces and Ramsey-classification theorems

We extend the hierarchy of finite-dimensional Ellentuck spaces to infinite dimensions. Using uniform barriers [Formula: see text] on [Formula: see text] as the prototype structures, we construct a class of continuum many topological Ramsey spaces [Formula: see text] which are Ellentuck-like in nature, and form a linearly ordered hierarchy under projections. We prove new Ramsey-classification theorems for equivalence relations on fronts, and hence also on barriers, on the spaces [Formula: see text], extending the Pudlák–Rödl theorem for barriers on the Ellentuck space. The inspiration for these spaces comes from continuing the iterative construction of the forcings [Formula: see text] to the countable transfinite. The [Formula: see text]-closed partial order [Formula: see text] is forcing equivalent to [Formula: see text], which forces a non-p-point ultrafilter [Formula: see text]. This work forms the basis for further work classifying the Rudin–Keisler and Tukey structures for the hierarchy of the generic ultrafilters [Formula: see text].

HIGH DIMENSIONAL ELLENTUCK SPACES AND INITIAL CHAINS IN THE TUKEY STRUCTURE OF NON-P-POINTS

AbstractThe generic ultrafilter${\cal G}_2 $forced by${\cal P}\left( {\omega \times \omega } \right)/\left( {{\rm{Fin}} \otimes {\rm{Fin}}} \right)$was recently proved to be neither maximum nor minimum in the Tukey order of ultrafilters ([1]), but it was left open where exactly in the Tukey order it lies. We prove${\cal G}_2 $that is in fact Tukey minimal over its projected Ramsey ultrafilter. Furthermore, we prove that for each${\cal G}_2 $, the collection of all nonprincipal ultrafilters Tukey reducible to the generic ultrafilter${\cal G}_k $forced by${\cal P}\left( {\omega ^k } \right)/{\rm{Fin}}^{ \otimes k} $forms a chain of lengthk. Essential to the proof is the extraction of a dense subsetεkfrom (Fin⊗k)+which we prove to be a topological Ramsey space. The spacesεk,k≥ 2, form a hierarchy of high dimensional Ellentuck spaces. New Ramsey-classification theorems for equivalence relations on fronts on εkare proved, extending the Pudlák–Rödl Theorem for fronts on the Ellentuck space, which are applied to find the Tukey and Rudin–Keisler structures below${\cal G}_k $.

Selective but not Ramsey

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 well-known 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.We show that for the topological Ramsey space R1 from [2], the notions of selective for R1 and Ramsey for R1 are not equivalent. In particular, we prove that forcing with a closely related topological Ramsey space using almost-reduction, adjoins an ultrafilter that is selective but not Ramsey for R1.

Ramsey for ultrafilter mappings and their Dedekind cuts

Associated to each ultrafilter on ω and each map is a Dedekind cut in the ultrapower . Blass has characterized, under , the cuts obtainable when 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 . Associated to the space is a notion of Ramsey ultrafilter for generalizing the familiar notion of Ramsey ultrafilter on ω. We characterize, under , the cuts obtainable when is taken to be a Ramsey for 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 using almost‐reduction adjoins an ultrafilter which is Ramsey for . For such ultrafilters , Dobrinen and Todorčević have shown that the Rudin‐Keisler types of the p‐points within the Tukey type of 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 the only cut obtainable is the standard cut. These results imply existence theorems for special kinds of ultrafilters.

Topological Ramsey spaces and metrically Baire sets

We characterize a class of topological Ramsey spaces such that each element R of the class induces a collection {Rk}k<ω of projected spaces which have the property that every Baire set is Ramsey. Every projected space Rk is a subspace of the corresponding space of length-k approximation sequences with the Tychonoff, equivalently metric, topology. This answers a question of S. Todorcevic and generalizes some results of Carlson, Carlson–Simpson, Prömel–Voigt, and Voigt. We also present a new family of topological Ramsey spaces contained in the aforementioned class which generalize the spaces of ascending parameter words of Carlson–Simpson and Prömel–Voigt and the spaces FINm[∞], 0<m<ω, of block sequences defined by Todorcevic.

A new class of Ramsey-Classification Theorems and their Applications in the Tukey Theory of Ultrafilters, Parts 1 and 2

Abstract Motivated by Tukey classification problems, we develop a new hierarchy of topological Ramsey spaces R α , α ω 1 . These spaces form a natural hierarchy of complexity, R 0 being the Ellentuck space [Erik Ellentuck, A new proof that analytic sets are Ramsey, Journal of Symbolic Logic 39 (1974), 163–165], and for each α ω 1 , R α + 1 coming immediately after R α in complexity. Associated with each R α is an ultrafilter U α , which is Ramsey for R α , and in particular, is a rapid p-point satisfying certain partition properties. We prove Ramsey-classification theorems for equivalence relations on fronts on R α , 1 ⩽ α ω 1 . These form a hierarchy of extensions of the Pudlak-Rodl Theorem canonizing equivalence relations on barriers on the Ellentuck space. We then apply our Ramsey-classification theorems to completely classify all Rudin-Keisler equivalence classes of ultrafilters which are Tukey reducible to U α , for each 1 ⩽ α ω 1 : Every nonprincipal ultrafilter which is Tukey reducible to U α is isomorphic to a countable iteration of Fubini products of ultrafilters from among a fixed countable collection of rapid p-points. Moreover, we show that the Tukey types of nonprincipal ultrafilters Tukey reducible to U α form a descending chain of rapid p-points of order type α + 1 .

A notion of selective ultrafilter corresponding to topological Ramsey spaces

AbstractWe 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[∞]. (© 2007 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)