Displaying similar documents to “Forcing with ideals generated by closed sets”

On nonmeasurable images

Robert Rałowski, Szymon Żeberski (2010)

Czechoslovak Mathematical Journal

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Let ( X , 𝕀 ) be a Polish ideal space and let T be any set. We show that under some conditions on a relation R ⊆ T 2 × X it is possible to find a set A ⊆ T such that R ( A 2 ) is completely 𝕀 -nonmeasurable, i.e, it is 𝕀 -nonmeasurable in every positive Borel set. We also obtain such a set A ⊆ T simultaneously for continuum many relations ( R α ) α < 2 ω . Our results generalize those from the papers of K. Ciesielski, H. Fejzić, C. Freiling and M. Kysiak.

On π -caliber and an application of Prikry’s partial order

Andrzej Szymański (2011)

Commentationes Mathematicae Universitatis Carolinae

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We study the concept of π -caliber as an alternative to the well known concept of caliber. π -caliber and caliber values coincide for regular cardinals greater than or equal to the Souslin number of a space. Unlike caliber, π -caliber may take on values below the Souslin number of a space. Under Martin’s axiom, 2 ω is a π -caliber of ℕ * . Prikry’s poset is used to settle a problem by Fedeli regarding possible values of very weak caliber.

Perfect sets and collapsing continuum

Miroslav Repický (2003)

Commentationes Mathematicae Universitatis Carolinae

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Under Martin’s axiom, collapsing of the continuum by Sacks forcing 𝕊 is characterized by the additivity of Marczewski’s ideal (see [4]). We show that the same characterization holds true if 𝔡 = 𝔠 proving that under this hypothesis there are no small uncountable maximal antichains in 𝕊 . We also construct a partition of ω 2 into 𝔠 perfect sets which is a maximal antichain in 𝕊 and show that s 0 -sets are exactly (subsets of) selectors of maximal antichains of perfect sets.

More results in polychromatic Ramsey theory

Uri Abraham, James Cummings (2012)

Open Mathematics

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We study polychromatic Ramsey theory with a focus on colourings of [ω 2]2. We show that in the absence of GCH there is a wide range of possibilities. In particular each of the following is consistent relative to the consistency of ZFC: (1) 2ω = ω 2 and ω 2 → p o l y ( α ) ℵ 0 - b d d 2 for every α <ω 2; (2) 2ω = ω 2 and ω 2 ↛ p o l y ( ω 1 ) 2 - b d d 2 .