Hostname: page-component-76fb5796d-vfjqv Total loading time: 0 Render date: 2024-04-25T14:05:54.473Z Has data issue: false hasContentIssue false
  • English
  • Français

GENERIC EXISTENCE OF MAD FAMILIES

Published online by Cambridge University Press:  21 March 2017

OSVALDO GUZMÁN-GONZÁLEZ ,
MICHAEL HRUŠÁK ,
CARLOS AZAREL MARTÍNEZ-RANERO  and
ULISES ARIET RAMOS-GARCÍA
OSVALDO GUZMÁN-GONZÁLEZ
Affiliation:
CENTRO DE CIENCIAS MATEMÁTICAS UNIVERSIDAD NACIONAL AUTÓNOMA DE MÉXICO CAMPUS MORELIA, MORELIA MICHOACÁN 58089, MÉXICOE-mail: oguzman@matmor.unam.mx
MICHAEL HRUŠÁK
Affiliation:
INSTITUTO DE MATEMÁTICAS UNIVERSIDAD NACIONAL AUTÓNOMA DE MÉXICO ÁREA DE LA INVESTIGACIÓN CIENTÍFICA CIRCUITO EXTERIOR, CIUDAD UNIVERSITARIA MÉXICO 04510, D.F., MÉXICOE-mail: michael@matmor.unam.mxURL: http://www.matmor.unam.mx/∼michael
CARLOS AZAREL MARTÍNEZ-RANERO
Affiliation:
DEPARTAMENTO DE MATEMÁTICA UNIVERSIDAD DE CONCEPCIÓN CASILLA 160-C, CONCEPCIÓN, CHILEE-mail: cmartinezr@udec.cl
ULISES ARIET RAMOS-GARCÍA
Affiliation:
CENTRO DE CIENCIAS MATEMÁTICAS UNIVERSIDAD NACIONAL AUTÓNOMA DE MÉXICO CAMPUS MORELIA, MORELIA MICHOACÁN 58089, MÉXICOE-mail: ariet@matmor.unam.mx
Get access Rights & Permissions [Opens in a new window]

Abstract

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.

Keywords

Primary 03E17Secondary 03E3503E40MAD familiescardinal invariants of the continuumforcing
Type
Articles
Information
The Journal of Symbolic Logic , Volume 82 , Issue 1 , March 2017 , pp. 303 - 316
Copyright
Copyright © The Association for Symbolic Logic 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Balcar, B., Hernández-Hernández, F., and Hrušák, M., Combinatorics of dense subsets of the rationals . Fundamenta Mathematicae, vol. 183 (2004), no. 1, pp. 59–80.Google Scholar
Blass, A., Combinatorial cardinal characteristics of the continuum , Handbook of Set Theory, (Foreman, M. and Kanamori, A., editors), Springer, Dordrecht, 2010, pp. 395–489.Google Scholar
Brendle, J., Around splitting and reaping . Commentationes Mathematicae Universitatis Carolinae, vol. 39 (1998), no. 2, pp. 269–279.Google Scholar
Brendle, J., The almost-disjointness number may have countable cofinality . Transactions of the American Mathematical Society, vol. 355 (2003), no. 7, pp. 2633–2649 (electronic).Google Scholar
Brendle, J. and Flašková, J., The generic existence of certain ${\cal I}$ -ultrafilters, preprint, 2011.Google Scholar
Brendle, J. and Yatabe, S., Forcing indestructibility of MAD families . Annals of Pure and Applied Logic, vol. 132 (2005), no. 2–3, pp. 271–312.Google Scholar
Canjar, R. M., On the generic existence of special ultrafilters . Proceedings of the American Mathematical Society, vol. 110 (1990), no. 1, pp. 233–241.Google Scholar
Fuchino, S., Geschke, S., and Soukup, L., How to drive our families mad, ArXiv Mathematics e-prints, 2006. arXiv:math/0611744 [math.LO].Google Scholar
Hrušák, M., Selectivity of almost disjoint families . Acta Universitatis Carolinae: Mathematica et Physica, vol. 41 (2000), no. 2, pp. 13–21.Google Scholar
Hrušák, M., MAD families and the rationals . Commentationes Mathematicae Universitatis Carolinae, vol. 42 (2001), no. 2, pp. 345–352.Google Scholar
Hrušák, M. and García Ferreira, S., Ordering MAD families a la Katétov, this Journal, vol. 68 (2003), no. 4, pp. 1337–1353.Google Scholar
Hrušák, M. and Zapletal, J., Forcing with quotients . Archive for Mathematical Logic, vol. 47 (2008), no. 7–8, pp. 719–739.Google Scholar
Kamburelis, A. and Węglorz, B., Splittings . Archive for Mathematical Logic, vol. 35 (1996), no. 4, pp. 263–277.Google Scholar
Kechris, A. S., Classical Descriptive Set Theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995.Google Scholar
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. 1583–1590.Google Scholar
Kunen, K., Ultrafilters and independent sets . Transactions of the American Mathematical Society, vol. 172 (1972), pp. 299–306.Google Scholar
Kuratowski, K., Introduction to Set Theory and Topology, PWN—Polish Scientific Publishers, Warsaw; Pergamon Press, Oxford-New York-Toronto, ON, 1977.Google Scholar
Kurilić, M. S., Cohen-stable families of subsets of integers, this Journal, vol. 66 (2001), no. 1, pp. 257–270.Google Scholar
Leathrum, T. E., A special class of almost disjoint families, this Journal, vol. 60 (1995), no. 3, pp. 879–891.Google Scholar
Malykhin, V. I., Topological properties of Cohen generic extensions . Trudy Moskovskogo Matematicheskogo Obshchestva, vol. 52 (1989), pp. 3–33, 247.Google Scholar
Mildenberger, H., Raghavan, D., and Steprans, J., Splitting families and complete separability . Canadian Mathematical Bulletin, vol. 57 (2014), no. 1, pp. 119–124.Google Scholar
Moore, J. T., Hrušák, M., and Džamonja, M., Parametrized ◊ principles . Transactions of the American Mathematical Society, vol. 356 (2004), no. 6, pp. 2281–2306.Google Scholar
Raghavan, D. and Steprāns, J., On weakly tight families . Canadian Journal of Mathematics, vol. 64 (2012), no. 6, pp. 1378–1394.Google Scholar
Shelah, S., MAD saturated families and SANE player . Canadian Journal of Mathematics, vol. 63 (2011), no. 6, pp. 1416–1435.Google Scholar
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. 583–617.Google Scholar
Zapletal, J., Forcing Idealized, Cambridge Tracts in Mathematics, vol. 174, Cambridge University Press, Cambridge, 2008.Google Scholar