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

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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

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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

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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

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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 .