Mad Spectra

The mad spectrum is the set of all cardinalities of infinite maximal almost disjoint families on omega. We treat the problem to characterize those sets A which, in some forcing extension of the universe, can be the mad spectrum. We solve this problem to some extent. What remains open is the possible values of min(A) and max(A)

Generic trees

We continue the investigation of the Laver ideal ℓ0 and Miller ideal m 0 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(m 0), 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 = ω 2 implies add(m 0) ≤ . We show that , implies cov(ℓ0) ≤ +, cov(m 0) ≤ + respectively. Here cov denotes the covering coefficient. We also show that in the Cohen model cov(m 0) < holds. Finally we prove that Cohen forcing does not add a superperfect tree of Cohen real

Dominating and unbounded free sets

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 projectiv

Independence and consistency proofs in quadratic form theory

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 theor