If is a filter on , we say that is Canjar if the corresponding Mathias forcing does not add a dominating real. We prove that any Borel Canjar filter is , solving a problem of Hrušák and Minami. We give several examples of Canjar and non-Canjar filters; in particular, we construct a 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 there are 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 on subsets of the Cantor space.
"Canjar Filters." Notre Dame J. Formal Logic 58 (1) 79 - 95, 2017. https://doi.org/10.1215/00294527-3496040