We prove that the Sacks forcing collapses the continuum onto ${\frak d}$ , answering the question of Carlson and Laver. Next we prove that if a proper forcing of the size at most continuum collapses $\omega_2$ then it forces $\diamondsuit_{\omega_{1}}$.

In this paper we study the question assuming MA+⌝CH does Sacks forcing or Laver forcing collapse cardinals? We show that this question is equivalent to the question of what is the additivity of Marczewski's ideals 0. We give a proof that it is consistent that Sacks forcing collapses cardinals. On the other hand we show that Laver forcing does not collapse cardinals.

Reviewed Works:John R. Steel, A. S. Kechris, D. A. Martin, Y. N. Moschovakis, Scales on $\Sigma^1_1$ Sets.Yiannis N. Moschovakis, Scales on Coinductive Sets.Donald A. Martin, John R. Steel, The Extent of Scales in $L$.John R. Steel, Scales in $L$.

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.

We continue [21] and study partition numbers of partial orderings which are related to /fin. In particular, we investigate Pf, be the suborder of /fin)ω containing only filtered elements, the Mathias partial order M, and , ω the lattice of partitions of ω, respectively. We show that Solomon's inequality holds for M and that it consistently fails for Pf. We show that the partition number of is C. We also show that consistently the distributivity number of ω is smaller than (...) the distributivity number of /fin. We also investigate partitions of a Polish space into closed sets. We show that such a partition either is countable or has size at least D, where D is the dominating number. We also show that the existence of a dominating family of size 1 does not imply that a Polish space can be partitioned into 1 many closed sets. (shrink)

We continue the investigation of the Laver ideal ℓ 0 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 , 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 (...) c = ω 2 implies $\operatorname{add}(m^0) \leq \mathfrak{h}$ . We show that b = c, d = c implies $\operatorname{cov}(\ell^0) \leq \mathfrak{h}^+, \operatorname{cov}(m^0) \leq \mathfrak{h}^+$ respectively. Here $\operatorname{cov}$ denotes the covering coefficient. We also show that in the Cohen model $\operatorname{cov}(m^0) holds. Finally we prove that Cohen forcing does not add a superperfect tree of Cohen reals. (shrink)

A family A ⊆ ℘(ω) is called countably splitting if for every countable $F \subseteq [\omega]^{\omega}$ , 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 $F_{\sigma}$ splitting family that does not contain a closed (...) splitting family. (shrink)