Displaying similar documents to “Sacks forcing collapses 𝔠 to 𝔟 ”

Partitions of k -branching trees and the reaping number of Boolean algebras

Claude Laflamme (1993)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

The reaping number 𝔯 m , n ( 𝔹 ) of a Boolean algebra 𝔹 is defined as the minimum size of a subset 𝒜 ⊆ 𝔹 ∖ { 𝐎 } such that for each m -partition 𝒫 of unity, some member of 𝒜 meets less than n elements of 𝒫 . We show that for each 𝔹 , 𝔯 m , n ( 𝔹 ) = 𝔯 ⌈ m n - 1 ⌉ , 2 ( 𝔹 ) as conjectured by Dow, Steprāns and Watson. The proof relies on a partition theorem for finite trees; namely that every k -branching tree whose maximal nodes are coloured with ℓ colours contains an m -branching subtree using at most n colours if and only if ⌈ ℓ n ⌉ < ⌈ k m - 1 ⌉ .

Pressing Down Lemma for λ -trees and its applications

Hui Li, Liang-Xue Peng (2013)

Czechoslovak Mathematical Journal

Similarity:

For any ordinal λ of uncountable cofinality, a λ -tree is a tree T of height λ such that | T α | < cf ( λ ) for each α < λ , where T α = { x ∈ T : ht ( x ) = α } . In this note we get a Pressing Down Lemma for λ -trees and discuss some of its applications. We show that if η is an uncountable ordinal and T is a Hausdorff tree of height η such that | T α | ≤ ω for each α < η , then the tree T is collectionwise Hausdorff if and only if for each antichain C ⊂ T and for each limit ordinal α ≤ η with cf ( α ) > ω , { ht ( c ) : c ∈ C } ∩ α is not stationary in α . In the last part of this note, we investigate...

OCA and towers in 𝒫 ( ℕ ) / f i n

Ilijas Farah (1996)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

We shall show that Open Coloring Axiom has different influence on the algebra 𝒫 ( ℕ ) / f i n than on ℕ ℕ / f i n . The tool used to accomplish this is forcing with a Suslin tree.

The tree property at the double successor of a measurable cardinal κ with 2 κ large

Sy-David Friedman, Ajdin Halilović (2013)

Fundamenta Mathematicae

Similarity:

Assuming the existence of a λ⁺-hypermeasurable cardinal κ, where λ is the first weakly compact cardinal above κ, we prove that, in some forcing extension, κ is still measurable, κ⁺⁺ has the tree property and 2 κ = κ ⁺ ⁺ ⁺ . If the assumption is strengthened to the existence of a θ -hypermeasurable cardinal (for an arbitrary cardinal θ > λ of cofinality greater than κ) then the proof can be generalized to get 2 κ = θ .

Fragments of strong compactness, families of partitions and ideal extensions

Laura Fontanella, Pierre Matet (2016)

Fundamenta Mathematicae

Similarity:

We investigate some natural combinatorial principles related to the notion of mild ineffability, and use them to obtain new characterizations of mild ineffable and weakly compact cardinals. We also show that one of these principles may be satisfied by a successor cardinal. Finally, we establish a version for κ ( λ ) of the canonical Ramsey theorem for pairs.

On ω 2 -saturated families

Lajos Soukup (1991)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

If there is no inner model with measurable cardinals, then for each cardinal λ there is an almost disjoint family 𝒜 λ of countable subsets of λ such that every subset of λ with order type ≥ ω 2 contains an element of 𝒜 λ .

On BPI Restricted to Boolean Algebras of Size Continuum

Eric Hall, Kyriakos Keremedis (2013)

Bulletin of the Polish Academy of Sciences. Mathematics

Similarity:

(i) The statement P(ω) = “every partition of ℝ has size ≤ |ℝ|” is equivalent to the proposition R(ω) = “for every subspace Y of the Tychonoff product 2 ( ω ) the restriction |Y = Y ∩ B: B ∈ of the standard clopen base of 2 ( ω ) to Y has size ≤ |(ω)|”. (ii) In ZF, P(ω) does not imply “every partition of (ω) has a choice set”. (iii) Under P(ω) the following two statements are equivalent: (a) For every Boolean algebra of size ≤ |ℝ| every filter can be extended to an ultrafilter. (b) Every Boolean...