Guzmán, Osvaldo; Hrušák, Michael; Martínez-Celis, Arturo Canjar filters. (English) Zbl 1417.03247 Notre Dame J. Formal Logic 58, No. 1, 79-95 (2017). Summary: 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 M. Hrušák and H. Minami [Ann. Pure Appl. Logic 165, No. 3, 880–894 (2014; Zbl 1306.03023)]. 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 J. Brendle [Arch. Math. Logic 37, No. 3, 183–197 (1998; Zbl 0905.03034)]. 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 \(S_{\mathrm{fin}}(\Omega,\Omega)\) on subsets of the Cantor space. Cited in 13 Documents MSC: 03E05 Other combinatorial set theory 03E17 Cardinal characteristics of the continuum 03E35 Consistency and independence results Keywords:Canjar filters; Mathias forcing; dominating reals; MAD families Citations:Zbl 1306.03023; Zbl 0905.03034 PDFBibTeX XMLCite \textit{O. Guzmán} et al., Notre Dame J. Formal Logic 58, No. 1, 79--95 (2017; Zbl 1417.03247) Full Text: DOI Link