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OCA and towers in 𝒫 ( ) / f i n

Ilijas Farah (1996)

Commentationes Mathematicae Universitatis Carolinae

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

On a problem of Steve Kalikow

Saharon Shelah (2000)

Fundamenta Mathematicae

The Kalikow problem for a pair (λ,κ) of cardinal numbers,λ > κ (in particular κ = 2) is whether we can map the family of ω-sequences from λ to the family of ω-sequences from κ in a very continuous manner. Namely, we demand that for η,ν ∈ ω we have: η, ν are almost equal if and only if their images are. We show consistency of the negative answer, e.g., for ω but we prove it for smaller cardinals. We indicate a close connection with the free subset property and its variants.

On absolutely divergent series

Sakaé Fuchino, Heike Mildenberger, Saharon Shelah, Peter Vojtáš (1999)

Fundamenta Mathematicae

We show that in the 2 -stage countable support iteration of Mathias forcing over a model of CH the complete Boolean algebra generated by absolutely divergent series under eventual dominance is not isomorphic to the completion of P(ω)/fin. This complements Vojtáš’ result that under c f ( ) = the two algebras are isomorphic [15].

On automorphisms of Boolean algebras embedded in P (ω)/fin

Magdalena Grzech (1996)

Fundamenta Mathematicae

We prove that, under CH, for each Boolean algebra A of cardinality at most the continuum there is an embedding of A into P(ω)/fin such that each automorphism of A can be extended to an automorphism of P(ω)/fin. We also describe a model of ZFC + MA(σ-linked) in which the continuum is arbitrarily large and the above assertion holds true.

On CCC boolean algebras and partial orders

András Hajnal, István Juhász, Zoltán Szentmiklóssy (1997)

Commentationes Mathematicae Universitatis Carolinae

We partially strengthen a result of Shelah from [Sh] by proving that if κ = κ ω and P is a CCC partial order with e.g. | P | κ + ω (the ω th successor of κ ) and | P | 2 κ then P is κ -linked.

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