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In this note we study generic existence of maximal almost disjoint (MAD) families. Among other results we prove that Cohen-indestructible families exist generically if and only if b = c. We obtain analogous results for other combinatorial properties of MAD families, including Sacks-indestructibility and being +-Ramsey.



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[1] Balcar, B., Hernández-Hernández, F., and Hrušák, M., Combinatorics of dense subsets of the rationals . Fundamenta Mathematicae, vol. 183 (2004), no. 1, pp. 5980.
[2] Blass, A., Combinatorial cardinal characteristics of the continuum , Handbook of Set Theory, (Foreman, M. and Kanamori, A., editors), Springer, Dordrecht, 2010, pp. 395489.
[3] Brendle, J., Around splitting and reaping . Commentationes Mathematicae Universitatis Carolinae, vol. 39 (1998), no. 2, pp. 269279.
[4] Brendle, J., The almost-disjointness number may have countable cofinality . Transactions of the American Mathematical Society, vol. 355 (2003), no. 7, pp. 26332649 (electronic).
[5] Brendle, J. and Flašková, J., The generic existence of certain ${\cal I}$ -ultrafilters, preprint, 2011.
[6] Brendle, J. and Yatabe, S., Forcing indestructibility of MAD families . Annals of Pure and Applied Logic, vol. 132 (2005), no. 2–3, pp. 271312.
[7] Canjar, R. M., On the generic existence of special ultrafilters . Proceedings of the American Mathematical Society, vol. 110 (1990), no. 1, pp. 233241.
[8] Fuchino, S., Geschke, S., and Soukup, L., How to drive our families mad, ArXiv Mathematics e-prints, 2006. arXiv:math/0611744 [math.LO].
[9] Hrušák, M., Selectivity of almost disjoint families . Acta Universitatis Carolinae: Mathematica et Physica, vol. 41 (2000), no. 2, pp. 1321.
[10] Hrušák, M., MAD families and the rationals . Commentationes Mathematicae Universitatis Carolinae, vol. 42 (2001), no. 2, pp. 345352.
[11] Hrušák, M. and García Ferreira, S., Ordering MAD families a la Katétov, this Journal, vol. 68 (2003), no. 4, pp. 13371353.
[12] Hrušák, M. and Zapletal, J., Forcing with quotients . Archive for Mathematical Logic, vol. 47 (2008), no. 7–8, pp. 719739.
[13] Kamburelis, A. and Węglorz, B., Splittings . Archive for Mathematical Logic, vol. 35 (1996), no. 4, pp. 263277.
[14] Kechris, A. S., Classical Descriptive Set Theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995.
[15] Keremedis, K., On the covering and the additivity number of the real line . Proceedings of the American Mathematical Society, vol. 123 (1995), no. 5, pp. 15831590.
[16] Kunen, K., Ultrafilters and independent sets . Transactions of the American Mathematical Society, vol. 172 (1972), pp. 299306.
[17] Kuratowski, K., Introduction to Set Theory and Topology, PWN—Polish Scientific Publishers, Warsaw; Pergamon Press, Oxford-New York-Toronto, ON, 1977.
[18] Kurilić, M. S., Cohen-stable families of subsets of integers, this Journal, vol. 66 (2001), no. 1, pp. 257270.
[19] Leathrum, T. E., A special class of almost disjoint families, this Journal, vol. 60 (1995), no. 3, pp. 879891.
[20] Malykhin, V. I., Topological properties of Cohen generic extensions . Trudy Moskovskogo Matematicheskogo Obshchestva, vol. 52 (1989), pp. 333, 247.
[21] Mildenberger, H., Raghavan, D., and Steprans, J., Splitting families and complete separability . Canadian Mathematical Bulletin, vol. 57 (2014), no. 1, pp. 119124.
[22] Moore, J. T., Hrušák, M., and Džamonja, M., Parametrizedprinciples . Transactions of the American Mathematical Society, vol. 356 (2004), no. 6, pp. 22812306.
[23] Raghavan, D. and Steprāns, J., On weakly tight families . Canadian Journal of Mathematics, vol. 64 (2012), no. 6, pp. 13781394.
[24] Shelah, S., MAD saturated families and SANE player . Canadian Journal of Mathematics, vol. 63 (2011), no. 6, pp. 14161435.
[25] Steprāns, J., Combinatorial consequences of adding Cohen reals , Set Theory of the Reals (Ramat Gan, 1991) (Judah, H., editor), Israel Mathematical Conference Proceedings, vol. 6, Bar-Ilan University, Ramat Gan, 1993, pp. 583617.
[26] Zapletal, J., Forcing Idealized, Cambridge Tracts in Mathematics, vol. 174, Cambridge University Press, Cambridge, 2008.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
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