Canjar filters

Abstract

If F is a filter on ϵ, 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 Fσ solving 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 Sfin.(ωω) on subsets of the Cantor space.