Abstract We show that in the Silver model the inequality $\mathrm {cov}(\mathfrak {C} _2) < \mathrm {cov}(\mathfrak {P}_2)$ holds true, where $\mathfrak {C}_2$ and $\mathfrak {P}_2$ are the two-dimensional Mycielski ideals.

Abstract The mad spectrum is the set of all cardinalities of infinite maximal almost disjoint families on ω. We treat the problem to characterize those sets ${\rm {\cal A}} $ which, in some forcing extension of the universe, can be the mad spectrum. We give a complete solution to this problem under the assumption $\vartheta ^{ < \vartheta } = \vartheta $ , where $\vartheta = {\rm{min}}\left( {\rm {\cal A}} \right) $ .

Abstract In this paper we investigate the structure of uncountable maximal antichains of Silver forcing and show that they have to be at least of size d, where d is the dominating number. Part of this work can be used to show that the additivity of the Silver forcing ideal has size at least the unbounding number b. It follows that every reasonable amoeba Silver forcing adds a dominating real.

Abstract A family A ⊆ (ω) is called countably splitting if for every countable F ⊆ [ω]ω, some element of A splits every member of F. We define a notion of a splitting tree, by means of which we prove that every analytic countably splitting family contains a closed countably splitting family. An application of this notion solves a problem of Blass. On the other hand we show that there exists an Fσ splitting family that does not contain a closed splitting family.

Abstract We prove that every analytic set in ωω × ωω with σ-bounded sections has a not σ-bounded closed free set. We show that this result is sharp. There exists a closed set with bounded sections which has no dominating analytic free set. and there exists a closed set with non-dominating sections which does not have a not σ-bounded analytic free set. Under projective determinacy analytic can be replaced in the above results by projective.

Abstract Two countable filters on ω are incompatible if they have no common infinite pseudointersection. Letting (Pf) denote the minimal size of a maximal uncountable family of pairwise incompatible countable filters on ω, we prove the consistency of t < a(Pf).

Abstract We continue the investigation of the Laver ideal ℓ0 and Miller ideal m0 started in [GJSp] and [GRShSp]; these are the ideals on the Baire space associated with Laver forcing and Miller forcing. We solve several open problems from these papers. The main result is the construction of models for t < add(ℓ0), < add(m0), where add denotes the additivity coefficient of an ideal. For this we construct amoeba forcings for these forcings which do not add Cohen reals. We show that = ω2 implies add(m0) ≤ . We show that , implies cov(ℓ0) ≤ +, cov(m0) ≤ + respectively. Here cov denotes the covering coefficient. We also show that in the Cohen model cov(m0) < holds. Finally we prove that Cohen forcing does not add a superperfect tree of Cohen reals.

We consider the following properties of uncountable-dimensional quadratic spaces (E, Φ): (*) For all subspaces U ⊆ E of infinite dimension: dim U˔ < dim E. (**) For all subspaces U ⊆ E of infinite dimension: dim U˔ < ℵ0. Spaces of countable dimension are the orthogonal sum of straight lines and planes, so they cannot have (*), but (**) is trivially satisfied. These properties have been considered first in [G/O] in the process of investigating the orthogonal group of quadratic spaces. It has been shown there (in ZFC) that over arbitrary uncountable fields (**)-spaces of uncountable dimension exist. In [B/G], (**)-spaces of dimension ℵ1 (so (*) = (**)) have been constructed over arbitrary finite or countable fields. But this could be done only under the assumption that the continuum hypothesis (CH) holds in the underlying set theory.