Abstract We study the values of the higher dimensional cardinal characteristics for sets of functions $f:\omega ^\omega \to \omega ^\omega $ introduced by the second author in [8]. We prove that while the bounding numbers for these cardinals can be strictly less than the continuum, the dominating numbers cannot. We compute the bounding numbers for the higher dimensional relations in many well known models of $\neg \mathsf {CH}$ such as the Cohen, random and Sacks models and, as a byproduct show that, with one exception, for the bounding numbers there are no $\mathsf {ZFC}$ relations between them beyond those in the higher dimensional Cichoń diagram. In the case of the dominating numbers we show that in fact they collapse in the sense that modding out by the ideal does not change their values. Moreover, they are closely related to the dominating numbers $\mathfrak {d}^\lambda _\kappa $ .

Abstract We investigate the structure of ultrafilters on Boolean algebras in the framework of Tukey reducibility. In particular, this paper provides several techniques to construct ultrafilters which are not Tukey maximal. Furthermore, we connect this analysis with a cardinal invariant of Boolean algebras, the ultrafilter number, and prove consistency results concerning its possible values on Cohen and random algebras.

Abstract We investigate which filters onωcan contain towers, that is, a modulo finite descending sequence without any pseudointersection (in ${[\omega ]^\omega }$ ). We prove the following results: (1) Many classical examples of nice tall filters contain no towers (in ZFC). (2) It is consistent that tall analytic P-filters contain towers of arbitrary regular height (simultaneously for many regular cardinals as well). (3) It is consistent that all towers generate nonmeager filters (this answers a question of P. Borodulin-Nadzieja and D. Chodounský), in particular (consistently) Borel filters do not contain towers. (4) The statement “Every ultrafilter contains towers.” is independent of ZFC (this improves an older result of K. Kunen, J. van Mill, and C. F. Mills). Furthermore, we study many possible logical (non)implications between the existence of towers in filters, inequalities between cardinal invariants of filters ( ${\rm{ad}}{{\rm{d}}^{\rm{*}}}\left( {\cal F} \right)$ , ${\rm{co}}{{\rm{f}}^{\rm{*}}}\left( {\cal F} \right)$ , ${\rm{no}}{{\rm{n}}^{\rm{*}}}\left( {\cal F} \right)$ , and ${\rm{co}}{{\rm{v}}^{\rm{*}}}\left( {\cal F} \right)$ ), and the existence of Luzin type families (of size $\ge {\omega _2}$ ), that is, if ${\cal F}$ is a filter then ${\cal X} \subseteq {[\omega ]^\omega }$ is an ${\cal F}$ -Luzin family if $\left\{ {X \in {\cal X}:|X \setminus F| = \omega } \right\}$ is countable for every $F \in {\cal F}$ .

Abstract We prove the consistency of together with the existence of a -definable mad family, answering a question posed by Friedman and Zdomskyy in [7, Question 16]. For the proof we construct a mad family in L which is an ℵ1-union of perfect a.d. sets, such that this union remains mad in the iterated Hechler extension. The construction also leads us to isolate a new cardinal invariant, the Borel almost-disjointness number , defined as the least number of Borel a.d. sets whose union is a mad family. Our proof yields the consistency of (and hence, ).

Abstract Let κ < λ be regular uncountable cardinals. Using a finite support iteration (in fact a matrix iteration) of ccc posets we obtain the consistency of . If μ is a measurable cardinal and μ < κ < λ, then using similar techniques we obtain the consistency of .

Abstract It is relatively consistent with ZFC that every countable FUfin space of weight ℵ1 is metrizable. This provides a partial answer to a question of G. Gruenhage and P. Szeptycki [GS1].

Abstract We investigate the set (ω) of partitions of the natural numbers ordered by ≤* where A ≤* B if by gluing finitely many blocks of A we can get a partition coarser than B. In particular, we determine the values of a number of cardinals which are naturally associated with the structure ((ω), ≥*), in terms of classical cardinal invariants of the continuum.

Abstract We show that g ≤ c(Sym(ω)) where g is the groupwise density number and c(Sym(ω)) is the cofinality of the infinite symmetric group. This solves (the second half of) a problem addressed by Thomas.

Abstract We give characterizations for the (in ZFC unprovable) sentences “Every -set is measurable” and “Every -set is measurable” for various notions of measurability derived from well-known forcing partial orderings.

Abstract We show that -Amoeba-absoluteness implies that and, hence, measurability. This answers a question of Haim Judah (private communication).

Abstract We prove that in many situations it is consistent with ZFC that part of the invariants involved in Cichoń's diagram are equal to κ while the others are equal to λ, where κ < λ are both arbitrary regular uncountable cardinals. We extend some of these results to the case when λ is singular. We also show that cf is consistent with ZFC.