Abstract We continue investigating variants of the splitting and reaping numbers introduced in [4]. In particular, answering a question raised there, we prove the consistency of and of . Moreover, we discuss their natural generalisations $\mathfrak {s}_{\rho }$ and $\mathfrak {r}_{\rho }$ for $\rho \in (0,1)$ , and show that $\mathfrak {r}_{\rho }$ does not depend on $\rho $ .

Abstract We study the following natural strong variant of destroying Borel ideals: $\mathbb {P}$ $+$ -destroys $\mathcal {I}$ if $\mathbb {P}$ adds an $\mathcal {I}$ -positive set which has finite intersection with every $A\in \mathcal {I}\cap V$ . Also, we discuss the associated variants $$ \begin{align*} \mathrm{non}^*(\mathcal{I},+)=&\min\big\{|\mathcal{Y}|:\mathcal{Y}\subseteq\mathcal{I}^+,\; \forall\;A\in\mathcal{I}\;\exists\;Y\in\mathcal{Y}\;|A\cap Y|<\omega\big\},\\ \mathrm{cov}^*(\mathcal{I},+)=&\min\big\{|\mathcal{C}|:\mathcal{C}\subseteq\mathcal{I},\; \forall\;Y\in\mathcal{I}^+\;\exists\;C\in\mathcal{C}\;|Y\cap C|=\omega\big\} \end{align*} $$ of the star-uniformity and the star-covering numbers of these ideals. Among other results, (1) we give a simple combinatorial characterisation when a real forcing $\mathbb {P}_I$ can $+$ -destroy a Borel ideal $\mathcal {J}$ ; (2) we discuss many classical examples of Borel ideals, their $+$ -destructibility, and cardinal invariants; (3) we show that the Mathias–Prikry, $\mathbb {M}(\mathcal {I}^*)$ -generic real $+$ -destroys $\mathcal {I}$ iff $\mathbb {M}(\mathcal {I}^*)\ +$ -destroys $\mathcal {I}$ iff $\mathcal {I}$ can be $+$ -destroyed iff $\mathrm {cov}^*(\mathcal {I},+)>\omega $ ; (4) we characterise when the Laver–Prikry, $\mathbb {L}(\mathcal {I}^*)$ -generic real $+$ -destroys $\mathcal {I}$ , and in the case of P-ideals, when exactly $\mathbb {L}(\mathcal {I}^*)$ $+$ -destroys $\mathcal {I}$ ; and (5) we briefly discuss an even stronger form of destroying ideals closely related to the additivity of the null ideal.

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 investigate ideals of the form {A⊆ω: Σn∈Axnis unconditionally convergent} where (xn)n∈ωis a sequence in a Polish group or in a Banach space. If an ideal onωcan be seen in this form for some sequence inX, then we say that it is representable inX. After numerous examples we show the following theorems: (1) An ideal is representable in a Polish Abelian group iff it is an analytic P-ideal. (2) An ideal is representable in a Banach space iff it is a nonpathological analytic P-ideal. We focus on the family of ideals representable inc0. We characterize this property via the defining sequence of measures. We prove that the trace of the null ideal, Farah’s ideal, and Tsirelson ideals are not representable inc0, and that a tallFσP-ideal is representable inc0iff it is a summable ideal. Also, we provide an example of a peculiar ideal which is representable inℓ1but not in ℝ. Finally, we summarize some open problems of this topic.

Abstract We prove the following version of Hechler's classical theorem: For each partially ordered set (Q, ≤) with the property that every countable subset of Q has a strict upper bound in Q, there is a ccc forcing notion such that in the generic extension for each tall analytic P-ideal (coded in the ground model) a cofinal subset of is order isomorphic to (Q, ≤).