Otmar Spinas

Dominating and Unbounded Free Sets

We prove that every analytic set in ω ω × ω ω 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.

Towers on trees

We show that (under MA) for any c many dense sets in Laver forcing L there exists a a-centered Q C L such that all the given dense sets are dense in Q. In particular, MA implies that L satisfies MA and does not collapse the continuum and the additivity of the Laver ideal is the continuum. The same is true for Miller forcing and for Mathias forcing. In the case of Miller forcing this involves the correction of the wrong proof of Judah, Miller, and Shelah, Sacks, Laver forcing, and Martins Axiom, Arch. Math. Logic 31 (1992), Theorem 4.1, p. 157.

The distributivity numbers of ()/fin and its square

We show that in a model obtained by forcing with a countable support iteration of Mathias forcing of length ω2, the distributivity number of P(ω)/fin is ω2, whereas the distributivity number of r.o.(P(ω)/fin) 2 is ω1. This answers an old problem of Balcar, Pelant and Simon, and others.

The essentially free spectrum of a variety

We partially prove a conjecture from [MeSh] which says that the spectrum of almost free, essentially free non-free algebras in a variety is either empty or consists of the class of all successor cardinals.

On tightness and depth in superatomic Boolean algebras

We introduce a large cardinal property which is consistent with L and show that for every superatomic Boolean algebra B and every cardinal λ with the large cardinal property, if tightness+(B) ≥ λ+, then depth(B) ≥ λ. This improves a theorem of Dow and Monk. In [DM, Theorem C], Dow and Monk have shown that if λ is a Ramsey cardinal (see [J, p.328]), then every superatomic Boolean algebra with tightness at least λ has depth at least λ. Recall that a Boolean algebra B is superatomic iff every homomorphic image of B is atomic. The depth of B is the supremum of all cardinals λ such that there is a sequence (bα : α < λ) in B with bβ < bα for all α < β < λ (a well-ordered chain of length λ). Then depth of B is the first cardinal λ such that there is no well-ordered chain of length λ in B. The tightness of B is the supremum of all cardinals λ such that B has a free sequence of length λ, where a sequence (bα : α < λ) is called free provided that if Γ and ∆ are finite subsets of λ such that α < β for all α ∈ Γ and β ∈ ∆, then ⋂

On the Baire Number of the Stone Space of P(ω)/fin

We prove that it is consistent with the axioms of ZFC, relative to the consistency of ZFC itself, that the Baire number of the Stone representation space of the Boolean algebra [Pscr ](ω)/fin equals the distributivity number of [Pscr ](ω)/fin. This answers a question of Balcar, Pelant and Simon in [ 1 ].