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)

This article is devoted to the interplay between forcing with fusion and combinatorial covering properties. We illustrate this interplay by proving that in the Laver model for the consistency of the Borel’s conjecture, the product of any two metrizable spaces with the Hurewicz property has the Menger property.

A maximal almost disjoint (mad) family $\mathscr{A} \subseteq [\omega]^\omega$ is Cohen-stable if and only if it remains maximal in any Cohen generic extension. Otherwise it is Cohen-unstable. It is shown that a mad family, A, is Cohen-unstable if and only if there is a bijection G from ω to the rationals such that the sets G[A], A ∈A are nowhere dense. An ℵ 0 -mad family, A, is a mad family with the property that given any countable family $\mathscr{B} \subset (...) [\omega]^\omega$ such that each element of B meets infinitely many elements of A in an infinite set there is an element of A meeting each element of B in an infinite set. It is shown that Cohen-stable mad families exist if and only if there exist ℵ 0 -mad families. Either of the conditions b = c or $\mathfrak{a} ) implies that there exist Cohen-stable mad families. Similar results are obtained for splitting families. For example, a splitting family, S, is Cohen-unstable if and only if there is a bijection G from ω to the rationals such that the boundaries of the sets G[S], S ∈S are nowhere dense. Also, Cohen-stable splitting families of cardinality ≤ κ exist if and only if ℵ 0 -splitting families of cardinality ≤ κ exist. (shrink)

We show the consistency of ${\frak o} <{\frak d}$ where ${\frak o}$ is the size of the smallest off-branch family, and ${\frak d}$ is as usual the dominating number. We also prove the consistency of ${\frak b} < {\frak a}$ with large continuum. Here, ${\frak b}$ is the unbounding number, and ${\frak a}$ is the almost disjointness number.