We present a survey of some results and problems concerning constructions which require a diagonalization of length continuum to be carried out, particularly constructions of almost disjoint families of various sorts. We emphasize the role of cardinal invariants of the continuum and their combinatorial characterizations in such constructions.

According to [M.S. Kurilić, Cohen-stable families of subsets of the integers, J. Symbolic Logic 66 257–270], adding a Cohen real destroys a splitting family on ω if and only if is isomorphic to a splitting family on the set of rationals, , whose elements have nowhere dense boundaries. Consequently, implies the Cohen-indestructibility of . Using the methods developed in [J. Brendle, S. Yatabe, Forcing indestructibility of MAD families, Ann. Pure Appl. Logic 132 271–312] the stability of splitting families in several (...) forcing extensions is characterized in a similar way . Also, it is proved that a splitting family is preserved by the Sacks forcing if and only if it is preserved by some forcing which adds a new real. The corresponding hierarchy of splitting families is investigated. (shrink)

A sequence x=xn:nω of elements of a complete Boolean algebra converges to a priori if lim infx=lim supx=b. The sequential topology τs on is the maximal topology on such that x→b implies x→τsb, where →τs denotes the convergence in the space — the a posteriori convergence. These two forms of convergence, as well as the properties of the sequential topology related to forcing, are investigated. So, the a posteriori convergence is described in terms of killing of tall ideals on ω, (...) and it is shown that the a posteriori convergence is equivalent to the a priori convergence iff forcing by does not produce new reals. A property of c.B.a.’s, satisfying -cc -cc and providing an explicit definition of the a posteriori convergence, is isolated. Finally, it is shown that, for an arbitrary c.B.a. , the space is sequentially compact iff the algebra has the property and does not produce independent reals by forcing, and that implies P is the unique sequentially compact c.B.a. in the class of Suslin forcing notions. (shrink)

We study an extensive connection between quotient forcings of Borel subsets of Polish spaces modulo a σ-ideal and quotient forcings of subsets of countable sets modulo an ideal.

We study the Katětov order on Borel ideals. We prove two structural theorems, one for Borel ideals, the other for analytic P-ideals. We isolate nine important Borel ideals and study the Katětov order among them. We also present a list of fundamental open problems concerning the Katětov order on Borel ideals.

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 $ ; (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. (shrink)

In this survey paper we collect several known results on destroying tall ideals on countable sets and maximal almost disjoint families with forcing. In most cases we provide streamlined proofs of the presented results. The paper contains results of many authors as well as a preview of results of a forthcoming paper of Brendle, Guzmán, Hrušák, and Raghavan.

Elekes proved that any infinite-fold cover of a σ-finite measure space by a sequence of measurable sets has a subsequence with the same property such that the set of indices of this subsequence has density zero. Applying this theorem he gave a new proof for the random-indestructibility of the density zero ideal. He asked about other variants of this theorem concerning I-almost everywhere infinite-fold covers of Polish spaces where I is a σ-ideal on the space and the set of indices (...) of the required subsequence should be in a fixed ideal ${{\mathcal{J}}}$ on ω. We introduce the notion of the ${{\mathcal{J}}}$ -covering property of a pair ${({\mathcal{A}}, I)}$ where ${{\mathcal{A}}}$ is a σ-algebra on a set X and ${{I \subseteq \mathcal{P}(X)}}$ is an ideal. We present some counterexamples, discuss the category case and the Fubini product of the null ideal ${\mathcal{N}}$ and the meager ideal ${\mathcal{M}}$ . We investigate connections between this property and forcing-indestructibility of ideals. We show that the family of all Borel ideals ${{\mathcal{J}}}$ on ω such that ${\mathcal{M}}$ has the ${{\mathcal{J}}}$ -covering property consists exactly of non weak Q-ideals. We also study the existence of smallest elements, with respect to Katětov–Blass order, in the family of those ideals ${\mathcal{J}}$ on ω such that ${\mathcal{N}}$ or ${\mathcal{M}}$ has the ${\mathcal{J}}$ -covering property. Furthermore, we prove a general result about the cases when the covering property “strongly” fails. (shrink)