If F is a filter on ω, we say that F is Canjar if the corresponding Mathias forcing does not add a dominating real. We prove that any Borel Canjar filter is F σ , this solves a problem of Hrušák and Minami. We give several examples of Canjar and non-Canjar filters, in particular, we construct a 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 ZFC there are 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 f in (Ω, Ω) on subsets of the Cantor space.

In this work we study cardinal invariants of the ideal SP of strongly porous sets on ω 2. We prove that add(SP) = ω 1 , cof(SP) = c and that it is consistent that non(SP) < add(N), answering questions of Hrušák and Zindulka. We also find a connection between strongly porous sets on ω 2 and the Martin number for σ-linked partial orders, and we use this connection to construct a model where all the Martin numbers for σ-k-linked forcings are mutually different.

Following D. Sobota we call a family F of infinite subsets of N a Rosenthal family if it can replace the family of all infinite subsets of N in classical Rosenthal's Lemma concerning sequences of measures on pairwise disjoint sets. We resolve two problems on Rosenthal families: every ultrafilter is a Rosenthal family and the minimal size of a Rosenthal family is exactly equal to the reaping cardinal r. This is achieved through analyzing nowhere reaping families of subsets of N and through applying a paving lemma which is a consequence of a paving lemma concerning linear operators on ℓ n 1 due to Bourgain. We use connections of the above results with free set results for functions on N and with linear operators on c 0 to determine the values of several other derived cardinal invariants.

Following D. Sobota we call a family F \mathcal F of infinite subsets of N \mathbb {N} a Rosenthal family if it can replace the family of all infinite subsets of N \mathbb {N} in the classical Rosenthal lemma concerning sequences of measures on pairwise disjoint sets. We resolve two problems on Rosenthal families: every ultrafilter is a Rosenthal family and the minimal size of a Rosenthal family is exactly equal to the reaping cardinal r \mathfrak r . This is achieved through analyzing nowhere reaping families of subsets of N \mathbb {N} and through applying a paving lemma which is a consequence of a paving lemma concerning linear operators on ℓ 1 n \ell _1^n due to Bourgain. We use connections of the above results with free set results for functions on N \mathbb {N} and with linear operators on c 0 c_0 to determine the values of several other derived cardinal invariants.