Adding ultrafilters by definable quotients

Hrušák, Michael; Verner, Jonathan L.

doi:10.1007/s12215-011-0064-0

Cited by 13 publications

(5 citation statements)

References 11 publications

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“…In particular it can not be a summable ideal. In [9] it has been proved that Hausdorff ultrafilters are exactly the -ultrafilters, which together with Theorem 7.1 imply that any Lafflame–Zhu ideal can not be Katětov–Blass above the ideal , the ideal on of graphs with finite chromatic number.…”

Section: Final Remarksmentioning

confidence: 99%

-Ultrafilters in the Rational Perfect Set Model

Cancino-Manríquez¹

2022

J. symb. log.

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We give a new characterization of the cardinal invariant $\mathfrak {d}$ as the minimal cardinality of a family $\mathcal {D}$ of tall summable ideals such that an ultrafilter is rapid if and only if it has non-empty intersection with all the ideals in the family $\mathcal {D}$ . On the other hand, we prove that in the Miller model, given any family $\mathcal {D}$ of analytic tall p-ideals such that $\vert \mathcal {D}\vert <\mathfrak {d}$ , there is an ultrafilter $\mathcal {U}$ which is an $\mathscr {I}$ -ultrafilter for all ideals $\mathscr {I}\in \mathcal {D}$ at the same time, yet $\mathcal {U}$ is not a rapid ultrafilter. As a corollary, we obtain that in the Miller model, given any analytic tall p-ideal $\mathscr {I}$ , $\mathscr {I}$ -ultrafilters are dense in the Rudin–Blass ordering, generalizing a theorem of Bartoszyński and S. Shelah, who proved that in such model, Hausdorff ultrafilters are dense in the Rudin–Blass ordering. This theorem also shows some limitations about possible generalizations of a theorem of C. Laflamme and J. Zhu.

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Section: Final Remarksmentioning

confidence: 99%

-Ultrafilters in the Rational Perfect Set Model

Cancino-Manríquez¹

2022

J. symb. log.

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“…Furthermore, the projection of P(ω × ω)/Fin ⊗2 to its first coordinate recovers P(ω)/Fin, and this projection of members of G 2 recovers a Ramsey ultrafilter on ω. Properties of this ultrafilter G 2 have been studied in [34], [20], [5], and [9]. The only non-principal ultrafilter Rudin-Keisler strictly below G 2 is exactly the Ramsey ultrafilter π 1 (G 2 ), or any ultrafilter isomorphic to it (Corollary 3.9 in [5]).…”

Section: A Hierarchy Of Non-p-points With Barren Extensionsmentioning

confidence: 99%

Classes of barren extensions

Dobrinen,

Hathaway

2019

Preprint

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Henle, Mathias, and Woodin proved in [19] that, provided that ω → (ω) ω holds in a model M of ZF, then forcing with ([ω] ω , ⊆ * ) 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 M [U ], where U is a Ramsey ultrafilter, with many properties of the original model M . This begged the question of how important the Ramseyness of U is for these results. In this paper, we show that several classes of σ-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 P(ω α )/Fin ⊗α , 2 ≤ α < ω 1 , forcing non-p-points also produce barren extensions.

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“…In [20], Hrušak and Verner proved that if I is a tall F σ P-ideal, then P(ω)/I adds a p-point which has no rapid RK-predessor and which is not Canjar. Thus, there is no Ramsey ultrafilter RK below this forced ultrafilter, but the Mathias forcing with tails in this ultrafilter does add a dominating real.…”

Section: Further Directionsmentioning

confidence: 99%

Topological Ramsey Spaces Dense in Forcings

Dobrinen¹

2017

Preprint

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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.

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Adding ultrafilters by definable quotients

Cited by 13 publications

References 11 publications

-Ultrafilters in the Rational Perfect Set Model

-Ultrafilters in the Rational Perfect Set Model

Classes of barren extensions

Topological Ramsey Spaces Dense in Forcings

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