Local Ramsey theory: an abstract approach

Prisco, Carlos Di; Mijares, José G.; Nieto, Jesús E.

doi:10.1002/malq.201500086

Cited by 13 publications

(21 citation statements)

References 26 publications

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“…In particular, our space E 2 provides an alternate method for proving Theorem 31 of [1], due to Blass, where it is shown that G 2 has the best partition properties that a non-p-point can have. We further remark that minor modifications to the arguments in [3] in conjunction with our theorems that E k is a topological Ramsey space and that P( k )/Fin ⊗k is forcing equivalent to (E k , ⊆ Fin ⊗k ) yield that these forcings P( )/Fin ⊗k have 'complete combinatorics' in the sense of Blass. The details will be included in a forthcoming paper of the author.…”

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confidence: 85%

“…The maximal nodes in Figure 1 is technically the set { i m : m < 20}, which indicates the structure of ↓≤3 . By Fin ⊗3 , we denote Fin ⊗ Fin ⊗2 , which consists of all subsets F ⊆ 3 such that for all but finitely many 3 with {(i, j, k) ∈ 3 : i < j < k}, we abuse notation and let Fin ⊗3 on [ ] 3 denote the collection of all subsets F ⊆ [ ] 3 such that {(i, j, k) : {i, j, k} ∈ F } is in Fin ⊗3 as defined on 3 . It is routine to check that (E 3 , ⊆ Fin ⊗3 ) is forcing equivalent to P( 3 )/Fin ⊗3 .…”

Section: A4 If Depth B (A) < ∞ and If O ⊆ Ar |A|+1 Then There Ismentioning

confidence: 99%

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High Dimensional Ellentuck Spaces and Initial Chains in the Tukey Structure of Non-P-Points

Dobrinen¹

2016

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The generic ultrafilter G 2 forced by P( × )/(Fin ⊗ Fin) 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 that G 2 is in fact Tukey minimal over its projected Ramsey ultrafilter. Furthermore, we prove that for each k ≥ 2, the collection of all nonprincipal ultrafilters Tukey reducible to the generic ultrafilter G k forced by P( k )/Fin ⊗k forms a chain of length k. Essential to the proof is the extraction of a dense subset E k from (Fin ⊗k ) + which we prove to be a topological Ramsey space. The spaces E k , k ≥ 2, form a hierarchy of high dimensional Ellentuck spaces. New Ramsey-classification theorems for equivalence relations on fronts on E k are proved, extending the Pudlák-Rödl Theorem for fronts on the Ellentuck space, which are applied to find the Tukey and Rudin-Keisler structures below G k . §1. Introduction. The structure of the Tukey types of ultrafilters is a current focus of research in set theory and structural Ramsey theory; the interplay between the two areas has proven fruitful for each. This particular line of research began in [14], in which Todorcevic showed that selective ultrafilters are minimal in the Tukey order via an insightful application of the Pudlák-Rödl Theorem canonizing equivalence relations on barriers on the Ellentuck space. Soon after, new topological Ramsey spaces were constructed by Dobrinen and Todorcevic in [7] and [8], in which Ramsey-classification theorems for equivalence relations on fronts were proved and applied to find initial Tukey structures of the associated p-point ultrafilters which are decreasing chains of order-type α + 1 for each countable ordinal α. Recent work of Dobrinen, Mijares, and Trujillo in [5] provided a template for constructing topological Ramsey spaces which have associated p-point ultrafilters with initial Tukey structures which are finite Boolean algebras, extending the work in [7]. This paper is the first to examine initial Tukey structures of non-p-points. Our work was motivated by [1], in which Blass, Dobrinen, and Raghavan studied the Tukey type of the generic ultrafilter G 2 forced by P( × )/Fin ⊗ Fin. As this ultrafilter was known to be a Rudin-Keisler immediate successor of its projected selective ultrafilter (see Proposition 30 in [1]) and at the same time be neither a p-point nor a Fubini iterate of p-points, it became of interest to see where in the Tukey hierarchy this ultrafilter lies.

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confidence: 85%

Section: A4 If Depth B (A) < ∞ and If O ⊆ Ar |A|+1 Then There Ismentioning

confidence: 99%

High Dimensional Ellentuck Spaces and Initial Chains in the Tukey Structure of Non-P-Points

Dobrinen¹

2016

J. symb. log.

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“…The third is R,≤ * where ≤ * is some -closed partial order which coarsens ≤ such that the separative quotients of R,≤ and R,≤ * are isomorphic. These forcings were shown to have many properties in common with Mathias forcing in [8,29]. Similarly to [ ] ,⊆ * , forcing with R,≤ * adds a new ultrafilter on a countable base set as follows: By Axiom A.2, relativizing below some member of R if necessary, the collection AR 1 of all first approximations to members of R is a countable set.…”

Section: Forcing With Topological Ramsey Spaces and The Ultrafilters They Generatementioning

confidence: 99%

“…So fix s ∈ [a,q] such that either [a,s] ġ ∈ S or [a,s] ġ ∈ S. Without loss of generality, assume the former.Since 1 is inaccessible in M, fix a g ∈ R that is P-generic over M such that g ∈ [a,s]. But P also has the Mathias property (see Theorem 6.24 in[8]). So every…”

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confidence: 99%

Classes of Barren Extensions

Dobrinen¹,

Hathaway²

2020

J. symb. log.

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Henle, Mathias, and Woodin proved in [21] that, provided that holds in a model M of ZF, then forcing with over M adds no new sets of ordinals, thus earning the name a “barren” extension. Moreover, under an additional assumption, they proved that this generic extension preserves all strong partition cardinals. This forcing thus produces a model , where is a Ramsey ultrafilter, with many properties of the original model M. This begged the question of how important the Ramseyness of is for these results. In this paper, we show that several classes of -closed forcings which generate non-Ramsey ultrafilters have the same properties. Such ultrafilters include Milliken–Taylor ultrafilters, a class of rapid p-points of Laflamme, k-arrow p-points of Baumgartner and Taylor, and extensions to a class of ultrafilters constructed by Dobrinen, Mijares, and Trujillo. Furthermore, the class of Boolean algebras , , forcing non-p-points also produce barren extensions.

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“…This question was answered positively in [2], and here we present an extension of this answer to the forcing localized on a semiselective coideal. In [3], this result was generalized to the corresponding notion analogous to Mathias forcing in a topological Ramsey space.…”

Section: Introductionmentioning

confidence: 99%

Ramsey subsets of the space of infinite block sequences of vectors

Calderón¹,

Prisco²,

Mijares³

2022

Fund. Math.

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We study families of infinite block sequences of elements of the space FIN k . In particular we study Ramsey properties of such families and Ramsey properties localized on a selective or semiselective coideal. We show how the stable ordered-union ultrafilters defined by Blass, and Matet-adequate families defined by Eisworth in the case k = 1, fit in the theory of the Ramsey space of infinite block sequences of finite sets of natural numbers.

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Local Ramsey theory: an abstract approach

Cited by 13 publications

References 26 publications

High Dimensional Ellentuck Spaces and Initial Chains in the Tukey Structure of Non-P-Points

High Dimensional Ellentuck Spaces and Initial Chains in the Tukey Structure of Non-P-Points

Classes of Barren Extensions

Ramsey subsets of the space of infinite block sequences of vectors

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