Partitioning pairs of countable sets of ordinals

1990 ◽  
Vol 55 (3) ◽  
pp. 1019-1021 ◽  
Author(s):  
Dan Velleman
Keyword(s):  
Partition Theorem ◽  
Partition Relations ◽  
Uncountable Cardinals ◽  
Image Position

In [3], Todorčević showed that ω1 ⇸ [ω1]ω12. In this paper we use similar methods to prove an analogous partition theorem for Pω1(λ), for certain uncountable cardinals λ.Recall that ω1 → [ω1]ω12, means that for every function f: [ω1]2 → ω1 there is a set A ∈ [ω1]ω1 such that f“[A]2 ≠ ω1, and of course ω1 ⇸ [ω1]ω12, is the negation of this statement. For partition relations on Pω1(→) it is customary to partition only those pairs of sets in which the first set is a subset of the second. Thus for A ⊆ Pω1(λ) we defineWe will write Pω1(λ) → [unbdd]λ2 to mean that for every function f: [Pω1(λ)]⊂2 → λ there is an unbounded set A ⊆ Pω1(λ) such that f“[A]⊂2 ≠ λ, and again Pω1(λ) ⇸ [unbdd]λ2 is the negation of this statement.

NEGATIVE PARTITION RELATIONS OR UNCOUNTABLE CARDINALS OF COFINALITY

2001 ◽  
Vol 37 (1-2) ◽  
pp. 233-236
Author(s):  
P. Matet
Keyword(s):  
Partition Relations ◽  
Uncountable Cardinals

We modify an argument of Baumgartner to show that…

On Finite Polarized Partition Relations

1969 ◽  
Vol 12 (3) ◽  
pp. 321-326 ◽  
Author(s):  
V. Chvátal
Keyword(s):  
Cardinal Numbers ◽  
Partition Relations ◽  
Image Position

Call an m × n array an m × n; k array if its mn entries come from a set of k elements. An m × n; 1 array has mn like entries. We write(1)if every m × n; k array contains a p × q; 1 sub-array. The negation of (1) is writtenand means that there is an m × n; k array containing no p × q; 1 sub-array. Relations (1) are called "polarized partition relations among cardinal numbers" by P. Erdös and R. Rado [2]. In this note we prove the following theorems.

scholarly journals UNIVERSAL CLASSES NEAR ${\aleph _1}$

2018 ◽  
Vol 83 (04) ◽  
pp. 1633-1643 ◽  
Author(s):  
MARCOS MAZARI-ARMIDA ◽  
SEBASTIEN VASEY
Keyword(s):  
Sufficient Conditions ◽  
List Type ◽  
Type Number ◽  
Abstract Elementary Classes ◽  
Uncountable Cardinal ◽  
Elementary Classes ◽  
Universal Classes ◽  
Uncountable Cardinals ◽  
Image Position

AbstractShelah has provided sufficient conditions for an ${\Bbb L}_{\omega _1 ,\omega } $-sentence ψ to have arbitrarily large models and for a Morley-like theorem to hold of ψ. These conditions involve structural and set-theoretic assumptions on all the ${\aleph _n}$’s. Using tools of Boney, Shelah, and the second author, we give assumptions on ${\aleph _0}$ and ${\aleph _1}$ which suffice when ψ is restricted to be universal:Theorem. Assume ${2^{{\aleph _0}}} < {2^{{\aleph _1}}}$. Let ψ be a universal ${\Bbb L}_{\omega _1 ,\omega } $-sentence.(1)If ψ is categorical in ${\aleph _0}$ and $1 \leqslant {\Bbb L}\left( {\psi ,\aleph _1 } \right) < 2^{\aleph _1 } $, then ψ has arbitrarily large models and categoricity of ψ in some uncountable cardinal implies categoricity of ψ in all uncountable cardinals.(2)If ψ is categorical in ${\aleph _1}$, then ψ is categorical in all uncountable cardinals.The theorem generalizes to the framework of ${\Bbb L}_{\omega _1 ,\omega } $-definable tame abstract elementary classes with primes.

Forcing many positive polarized partition relations between a cardinal and its powerset

2001 ◽  
Vol 66 (3) ◽  
pp. 1359-1370 ◽  
Author(s):  
Saharon Shelah ◽  
Lee J. Stanley
Keyword(s):  
Natural Number ◽  
Representative Case ◽  
Partition Relations ◽  
Image Position

AbstractA fairly quotable special, but still representative, case of our main result is that for 2 ≤ n < ω, there is a natural number m(n) such that, the following holds. Assume GCH: If λ < μ are regular, there is a cofinality preserving forcing extension in which 2λ = μ and, for all σ < λ ≤ κ < η such that η(+m(n)−+) ≤ μ,This generalizes results of [3], Section 1. and the forcing is a “many cardinals” version of the forcing there.

THE COUNTERPARTS TO STATEMENTS THAT ARE EQUIVALENT TO THE CONTINUUM HYPOTHESIS

2015 ◽  
Vol 80 (4) ◽  
pp. 1075-1090
Author(s):  
ASGER TÖRNQUIST ◽  
WILLIAM WEISS
Keyword(s):  
Continuum Hypothesis ◽  
Partition Relations ◽  
Image Position ◽  
The Continuum

AbstractWe consider natural ${\rm{\Sigma }}_2^1$ definable analogues of many of the classical statements that have been shown to be equivalent to CH. It is shown that these ${\rm{\Sigma }}_2^1$ analogues are equivalent to that all reals are constructible. We also prove two partition relations for ${\rm{\Sigma }}_2^1$ colourings which hold precisely when there is a non-constructible real.

THE NEXT BEST THING TO A P-POINT

2015 ◽  
Vol 80 (3) ◽  
pp. 866-900 ◽  
Author(s):  
ANDREAS BLASS ◽  
NATASHA DOBRINEN ◽  
DILIP RAGHAVAN
Keyword(s):  
Isomorphism Class ◽  
Partition Relations ◽  
Image Position ◽  
Selective Ultrafilter

AbstractWe study ultrafilters on ω2 produced by forcing with the quotient of ${\cal P}$(ω2) by the Fubini square of the Fréchet filter on ω. We show that such an ultrafilter is a weak P-point but not a P-point and that the only nonprincipal ultrafilters strictly below it in the Rudin–Keisler order are a single isomorphism class of selective ultrafilters. We further show that it enjoys the strongest square-bracket partition relations that are possible for a non-P-point. We show that it is not basically generated but that it shares with basically generated ultrafilters the property of not being at the top of the Tukey ordering. In fact, it is not Tukey-above [ω1]<ω, and it has only continuum many ultrafilters Tukey-below it. A tool in our proofs is the analysis of similar (but not the same) properties for ultrafilters obtained as the sum, over a selective ultrafilter, of nonisomorphic selective ultrafilters.

scholarly journals Strong sequences and partition relations

2017 ◽  
Vol 16 (1) ◽  
pp. 51-59
Author(s):  
Joanna Jureczko
Keyword(s):  
The Other ◽  
Generalize Theorem ◽  
Partition Theorem ◽  
Other Hand ◽  
Partition Relations

AbstractThe first result in partition relations topic belongs to Ramsey (1930). Since that this topic has been still explored. Probably the most famous partition theorem is Erdös-Rado theorem (1956). On the other hand in 60’s of the last century Efimov introduced strong sequences method, which was used for proving some famous theorems in dyadic spaces. The aim of this paper is to generalize theorem on strong sequences and to show that it is equivalent to generalized version of well-known Erdös-Rado theorem. It will be also shown that this equivalence holds for singulars. Some applications and conclusions will be presented too.

Rothberger's property and partition relations

1997 ◽  
Vol 62 (3) ◽  
pp. 976-980 ◽  
Author(s):  
Marion Scheepers
Keyword(s):  
Finite Subset ◽  
Metric Space ◽  
Open Cover ◽  
Partition Relation ◽  
Partition Relations ◽  
Image Position ◽  
Separable Metric Space

Let X be an infinite but separable metric space. An open cover of X is said to be large if for each x ϵ X the set {U ϵ : x ϵ U} is infinite. The symbol Λ denotes the collection of large open covers of X. An open cover of X is said to be an ω-cover if for each finite subset F of X there is a U ϵ such that F ⊆ U, and X is not a member of , X is said to have Rothberger's property if there is for every sequence (n : n = 1,2,3,…) of open covers of X a sequence (Un : n = 1,2,3,…) such that:(1) for each n, Un is a member of n, and(2) {Un: n = 1,2,3,…} is a cover of X.Rothberger introduced this property in his paper [2]. For convenience we let denote the collection of all open covers of X.In [3] it was shown that X has Rothberger's property if, and only if, the following partition relation is true for large open covers of X:This partition relation means:for every large cover of X, for every coloringsuch that for each U ϵ and each large cover there is an i with a large cover of X,either there is a large cover such that f({A, B}) = 0 whenever {A,B} ϵ ,or else there is a which is not point–finite such that f{{A, B}) = 1 whenever {A, B} ϵ .

scholarly journals UNCOUNTABLE TREES AND COHEN -REALS

2019 ◽  
Vol 84 (3) ◽  
pp. 877-894 ◽  
Author(s):  
GIORGIO LAGUZZI
Keyword(s):  
Strong Relationship ◽  
Standard Case ◽  
Regularity Properties ◽  
Uncountable Cardinals ◽  
Image Position

AbstractWe investigate some versions of amoeba for tree-forcings in the generalized Cantor and Baire spaces. This answers [10, Question 3.20] and generalizes a line of research that in the standard case has been studied in [11], [13], and [7]. Moreover, we also answer questions posed in [3] by Friedman, Khomskii, and Kulikov, about the relationships between regularity properties at uncountable cardinals. We show ${\bf{\Sigma }}_1^1$-counterexamples to some regularity properties related to trees without club splitting. In particular we prove a strong relationship between the Ramsey and the Baire properties, in slight contrast with the standard case.

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