Ultrafilters of character <i>ω</i><sub>1</sub>

Hart, Klaas Pieter

doi:10.2307/2275010

Cited by 5 publications

(12 citation statements)

References 8 publications

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“…Note also that similarly as for Remark 5.3.5 we can argue the following fact which, in a model obtained by adding many side-by-side Sacks reals, was first noticed by Hart in [58]. Recall also that a family J ⊂ [ω] ω is an independent family provided the set…”

Section: Fact 541 Ifsupporting

confidence: 63%

“…This fact was first noticed (in a model obtained by adding many side-by-side Sacks reals) by Hart in [58]. Proof.…”

Section: Fact 531 the Ideal I Of Nowhere Dense Subset Of Rationals mentioning

confidence: 60%

“…It is worth mentioning that CPA game prism implies also the existence of many other kinds of ultrafilters, like these constructed in [58]. In fact, many constructions that are done under CH can be carried also under CPA game prism .…”

Section: Crowded Ultrafilters On Qmentioning

confidence: 99%

“…4.4.7].) Hart in [58] proved that in a model obtained by adding many side-by-side Sacks reals there is a P -point which is not selective. Next, we will prove that this follows also from CPA game prism .…”

Section: Non-selective P -Points and Number Imentioning

confidence: 99%

“…Next, we will prove that this follows also from CPA game prism . The main idea of the proof is the same as that used in [58].…”

Section: Non-selective P -Points and Number Imentioning

confidence: 99%

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The Covering Property Axiom, CPA

Ciesielski

Pawlikowski

2004

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Section: Fact 541 Ifsupporting

confidence: 63%

“…This fact was first noticed (in a model obtained by adding many side-by-side Sacks reals) by Hart in [58]. Proof.…”

Section: Fact 531 the Ideal I Of Nowhere Dense Subset Of Rationals mentioning

confidence: 60%

Section: Crowded Ultrafilters On Qmentioning

confidence: 99%

Section: Non-selective P -Points and Number Imentioning

confidence: 99%

“…Next, we will prove that this follows also from CPA game prism . The main idea of the proof is the same as that used in [58].…”

Section: Non-selective P -Points and Number Imentioning

confidence: 99%

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The Covering Property Axiom, CPA

Ciesielski

Pawlikowski

2004

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Unfriendly partitions when avoiding vertices of finite degree

Fiorini Aurichi,

Silva Sinzato Real

2023

Journal of Logic and Computation

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An unfriendly partition of a graph $G = (V,E)$ is a function $c: V \to 2$ such that $|\{x\in N(v): c(x)\neq c(v)\}|\geq |\{x\in N(v): c(x)=c(v)\}|$ for every vertex $v\in V$, where $N(v)$ denotes its neighborhood. It was conjectured by Cowan and Emerson [2] that every graph has an unfriendly partition, but Milner and Shelah in [5] found counterexamples for that statement by analysing graphs with uncountably many vertices. Curiously, none of their graphs have vertices with finite degree. Therefore, as a natural direction to approach, in this paper we search for the least cardinality of a graph with this property and that admits no unfriendly partitions. Actually, among some other independence results, we conclude that this value cannot be precisely determined within $\textrm {ZFC}$, in the sense that it may vary from model to model of set theory.

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Crowded and Selective Ultrafilters under the Covering Property Axiom

Ciesielski¹,

Pawlikowski²

2003

Journal of Applied Analysis

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In the paper we formulate an axiom CPA game prism , which is the most prominent version of the Covering Property Axiom CPA, and discuss several of its implications. In particular, we show that it implies that the following cardinal characteristics of continuum are equal to ω 1 , while c = ω 2 : the independence number i, the reaping number r, the almost disjoint number a, and the ultrafilter base number u. We will also show that CPA game prism implies the existence of crowded and selective ultrafilters as well as nonselective P -points. In addition we prove that under CPA game prism every selective ultrafilter is ω 1 -generated. The paper finishes with the proof that CPA game prism holds in the iterated perfect set model. * AMS classification numbers: Primary 54D80, 03E17; Secondary 03E35.

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Ultrafilters of character ω₁

Abstract: Using side-by-side Sacks forcing, it is shown that it is consistent that 2ω be large and that there be many types of ultrafilters of character ω1.

Cited by 5 publications

References 8 publications

The Covering Property Axiom, CPA

The Covering Property Axiom, CPA

Unfriendly partitions when avoiding vertices of finite degree

Crowded and Selective Ultrafilters under the Covering Property Axiom

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Ultrafilters of character ω1

Abstract: Using side-by-side Sacks forcing, it is shown that it is consistent that 2ω be large and that there be many types of ultrafilters of character ω1.

Cited by 5 publications

References 8 publications

The Covering Property Axiom, CPA

The Covering Property Axiom, CPA

Unfriendly partitions when avoiding vertices of finite degree

Crowded and Selective Ultrafilters under the Covering Property Axiom

Contact Info

Product

Resources

About

Ultrafilters of character ω₁