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.

If $\mathcal{F}$ is a filter on $\omega$, we say that $\mathcal{F}$ is Canjar if the corresponding Mathias forcing does not add a dominating real. We prove that any Borel Canjar filter is $F_{\sigma}$, solving a problem of Hrušák and Minami. We give several examples of Canjar and non-Canjar filters; in particular, we construct a $\mathsf{MAD}$ family such that the corresponding Mathias forcing adds a dominating real. This answers a question of Brendle. Then we prove that in all the “classical” models (...) of $\mathsf{ZFC}$ there are $\mathsf{MAD}$ families whose Mathias forcing does not add a dominating real. We also study ideals generated by branches, and we uncover a close relation between Canjar ideals and the selection principle $S_{\mathrm{fin}}$ on subsets of the Cantor space. (shrink)

We show that Hechler’s forcings for adding a tower and for adding a mad family can be represented as finite support iterations of Mathias forcings with respect to filters and that these filters are $${\mathcal {B}}$$ B -Canjar for any countably directed unbounded family $${\mathcal {B}}$$ B of the ground model. In particular, they preserve the unboundedness of any unbounded scale of the ground model. Moreover, we show that $${\mathfrak {b}}=\omega _1$$ b = ω 1 in every extension by the (...) above forcing notions. (shrink)

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)

We give topological characterizations of filters${\cal F}$onωsuch that the Mathias forcing${M_{\cal F}}$adds no dominating reals or preserves ground model unbounded families. This allows us to answer some questions of Brendle, Guzmán, Hrušák, Martínez, Minami, and Tsaban.

We study the Mathias–Prikry and the Laver type forcings associated with filters and coideals. We isolate a crucial combinatorial property of Mathias reals, and prove that Mathias–Prikry forcings with summable ideals are all mutually bi-embeddable. We show that Mathias forcing associated with the complement of an analytic ideal always adds a dominating real. We also characterize filters for which the associated Mathias–Prikry forcing does not add eventually different reals, and show that they are countably generated provided they are Borel. We (...) give a characterization of ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\omega}$$\end{document}-hitting and ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\omega}$$\end{document}-splitting families which retain their property in the extension by a Laver type forcing associated with a coideal. (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.

Let I be an F σ -ideal on natural numbers. We characterize the ultrafilters which are generic over the model L for the poset of I -positive sets of natural numbers ordered by inclusion.

We investigate which filters onωcan contain towers, that is, a modulo finite descending sequence without any pseudointersection. We prove the following results:Many classical examples of nice tall filters contain no towers.It is consistent that tall analytic P-filters contain towers of arbitrary regular height.It is consistent that all towers generate nonmeager filters, in particular Borel filters do not contain towers.The statement “Every ultrafilter contains towers.” is independent of ZFC.Furthermore, we study many possible logical implications between the existence of towers in filters, (...) inequalities between cardinal invariants of filters $,${\rm{co}}{{\rm{f}}^{\rm{*}}}\left$,${\rm{no}}{{\rm{n}}^{\rm{*}}}\left$, and${\rm{co}}{{\rm{v}}^{\rm{*}}}\left$), and the existence of Luzin type families, that is, if${\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}$. (shrink)

In this paper we discuss what kind of constrains combinatorial covering properties of Menger, Scheepers, and Hurewicz impose on remainders of topological groups. For instance, we show that such a remainder is Hurewicz if and only it is σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document}-compact. Also, the existence of a Scheepers non-σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document}-compact remainder of a topological group follows from CH and yields a P-point, and hence is (...) independent of ZFC. We also make an attempt to prove a dichotomy for the Menger property of remainders of topological groups in the style of Arhangel’skii. (shrink)