Selective but not Ramsey

Trujillo, Timothy

doi:10.1016/j.topol.2015.12.088

Cited by 9 publications

(14 citation statements)

References 15 publications

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“…In this section, we shall describe the main properties of the forcing notion (H, ≤ * ), for a semiselective H. To do so, we shall need to consider a special type of coideal U ⊆ R: Definition 6.1 Given U ⊆ R, we say that U is an ultrafilter if it satisfies the following: In [29], a similar abstract definition of ultrafilter is used. But the ultrafilters used in [29] do not satisfy part (c) of Definition 6.1.…”

Section: The Souslin Operationmentioning

confidence: 99%

Local Ramsey theory: an abstract approach

Prisco

Mijares

Nieto

2017

Mathematical Logic Qtrly

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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. ).

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Section: The Souslin Operationmentioning

confidence: 99%

Local Ramsey theory: an abstract approach

Prisco

Mijares

Nieto

2017

Mathematical Logic Qtrly

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“…However it is still unknown if, under

sans-serifCH

, there are selective for

R_{α}

ultrafilters on

[T_{α}]

which are not Ramsey for

R_{α}

for

ω \leq α < ω_{1}

. The methods used in fail for infinite α as the trees

T_{α}

have infinite height. Question Assume that

1 < α < ω_{1}

.…”

Section: Resultsmentioning

confidence: 99%

Ramsey for ultrafilter mappings and their Dedekind cuts

Trujillo

2015

Mathematical Logic Qtrly

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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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“…Let w ′ denote π r j (v) (v(j)). Then w ′ is the ≺-least member of ϕ ′′ (v) satisfying w ≺ w ′ , so by observation (16), we see that {π r l (v) (v(l)) ≺ w : l < |v|} = ϕ ′′ (r j (v)). Therefore, ϕ ′′ (r i (u)) = ϕ ′′ (r j (v)), since w is the ≺-least member of ϕ ′′ (u)△ϕ ′′ (v).…”

Section: Proofmentioning

confidence: 96%

“…Next, we show that ϕ holds the same information about R that ϕ ′′ does. The following observation will be useful: For each u ∈F|W , (16) π r 0 (u) (u(0)) u(0) ≺ π r 1 (u) (u(1)) u(1) ≺ · · · .…”

Section: Proofmentioning

confidence: 99%

“…Similar to the connection between the Ellentuck space and Ramsey ultrafilters, each topological Ramsey space has associated ultrafilters, which are 'selective' or 'Ramsey' with respect to the space (see [15] and [16]). Using the Continuum Hypothesis or Martin's Axiom (indeed, less is actually required) or forcing, any topological Ramsey space gives rise to an associated ultrafilter satisfying some partition properties, dependent on the space.…”

Section: Introductionmentioning

confidence: 99%

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Infinite-dimensional Ellentuck spaces and Ramsey-classification theorems

Dobrinen

2016

J. Math. Log.

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We extend the hierarchy of finite-dimensional Ellentuck spaces to infinite dimensions. Using uniform barriers B on ω as the prototype structures, we construct a class of continuum many topological Ramsey spaces EB which are Ellentuck-like in nature, and form a linearly ordered hierarchy under projection. We prove new Ramseyclassification theorems for equivalence relations on fronts, and hence also on barriers, on the spaces EB, 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 P([ω] k )/Fin ⊗k to the countable transfinite. The σ-closed partial order (EB, ⊆ Fin B ) is forcing equivalent to P(B)/Fin B , which forces a non-p-point ultrafilter GB. The present work forms the basis for further work classifying the Rudin-Keisler and Tukey structures for the hierarchy of the generic ultrafilters GB.2010 Mathematics Subject Classification. 03E05, 03E02, 05D10.

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Selective but not Ramsey

Cited by 9 publications

References 15 publications

Local Ramsey theory: an abstract approach

Local Ramsey theory: an abstract approach

Ramsey for ultrafilter mappings and their Dedekind cuts

Infinite-dimensional Ellentuck spaces and Ramsey-classification theorems

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