Andrzej Roslanowski

Norms on Possibilities II: More CCC Ideals on 2 ω

Abstract We use the method of norms on possibilities to answer a question of Kunen and construct a ccc σ–ideal on 2 ω with various closure properties and distinct from the ideal of null sets, the ideal of meager sets and their intersection.

Reasonable Ultrafilters, Again

We continue investigations of reasonable ultrafilters on uncountable cardinals defined in math.LO/0407498. We introduce stronger properties of ultrafilters and we show that those properties may be handled in lambda-support iterations of reasonably bounding forcing notions. We use this to show that consistently there are reasonable ultrafilters on an inaccessible cardinal lambda with generating system of size less than 2^lambda . We also show how reasonable ultrafilters can be killed by forcing notions which have enough reasonable completeness to be iterated with lambda-supports (and we show the appropriate preservation theorem).

More on cardinal invariants of Boolean algebras

Abstract We address several questions of Donald Monk related to irredundance and spread of Boolean algebras, gaining both some ZFC knowledge and consistency results. We show in ZFC that irr ( B 0 × B 1 )= max { irr ( B 0 ), irr ( B 1 )} . We prove consistency of the statement “there is a Boolean algebra B such that irr ( B ) B ⊛ B ) ” and we force a superatomic Boolean algebra B ∗ such that s( B ∗ )= inc ( B ∗ )=κ , irr ( B ∗ )= Id ( B ∗ )=κ + and Sub ( B ∗ )=2 κ + . Next we force a superatomic algebra B 0 such that irr ( B 0 ) inc ( B 0 ) and a superatomic algebra B 1 such that t ( B 1 )> Aut ( B 1 ) . Finally we show that consistently there is a Boolean algebra B of size λ such that there is no free sequence in B of length λ, there is an ultrafilter of tightness λ (so t ( B )=λ ) and λ∉ Depth Hs ( B ) .

On thep-rank of Ext

AssumeV =L andλ is regular smaller than the first weakly compact cardinal. Under those circumstances and with arbitrary requirements on the structure of Ext (G, ℤ) (under well known limitations), we construct an abelian groupG of cardinalityλ such that for noG′ ⊆G, |G′| <λ isG/G′ free and Ext (G, ℤ) realizes our requirements.

Ideals without CCC

Let I be an ideal of subsets of a Polish space X, containing all singletons and possessing a Borel basis. Assuming that I does not satisfy ccc, we consider the following conditions (B), (M) and (D). Condition (B) states that there is a disjoint family F ⊆ P (X) of size ϲ, consisting of Borel sets which are not in I . Condition (M) states that there is a Borel function f : X → X with f −1 [{ x }] ∉ I for each x ∈ X . Provided that X is a group and I is invariant, condition (D) states that there exist a Borel set B ∉ I and a perfect set P ⊆ X for which the family { B + x : x ∈ P } is disjoint. The aim of the paper is to study whether the reverse implications in the chain (D) ⇒ (M) ⇒ (B) ⇒ not-ccc can hold. We build a σ-ideal on the Cantor group witnessing (M) & ¬(D) (Section 2). A modified version of that σ-ideal contains the whole space (Section 3). Some consistency results on deriving (M) from (B) for “nicely” defined ideals are established (Sections 4 and 5). We show that both ccc and (M) can fail (Theorems 1.3 and 5.6). Finally, some sharp versions of (M) for invariant ideals on Polish groups are investigated (Section 6).

n—localization property

Let n be an integer greater than 1. A tree T is an n-ary tree provided that every node in T has at most n immediate successors. A forcing notion P has the n-localization property if every function from omega to omega in an extension via P is an omega-branch in an n-ary tree from the ground model. In the present paper we are interested in getting the n-localization property for countable support iterations.

ADDING ONE RANDOM REAL

We study the cardinal invariants of measure and category after adding one random real. In particular, we show that the number of measure zero subsets of the plane which are necessary to cover graphs of all continuous functions maybe large while the covering for measure is small.

Small-large subgroups of the reals

Abstract We are interested in subgroups of the reals that are small in one and large in another sense. We prove that, in ZFC, there exists a non-meager Lebesgue null subgroup of ℝ, while it is consistent that there there is no non-null meager subgroup of ℝ.

How much sweetness is there in the universe

We continue investigations of forcing notions with strong ccc properties introducing new methods of building sweet forcing notions. We also show that quotients of topologically sweet forcing notions over Cohen reals are topologically sweet while the quotients over random reals do not have to be such. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Generating ultrafilters in a reasonable way

We continue investigations of reasonable ultrafilters on uncountable cardinals defined in Shelah [8]. We introduce a general scheme of generating a filter on λ from filters on smaller sets and we investigate the combinatorics of objects obtained this way. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)