Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

There is a Van Douwen MAD family


Author: Dilip Raghavan
Journal: Trans. Amer. Math. Soc. 362 (2010), 5879-5891
MSC (2010): Primary 03E17, 03E15, 03E05, 03E50
DOI: https://doi.org/10.1090/S0002-9947-2010-04975-X
Published electronically: June 11, 2010
MathSciNet review: 2661500
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We answer a long-standing question of Van Douwen by proving in $ \mathrm{ZFC}$ that there is a MAD family of functions in $ {\omega}^{\omega}$ that is also maximal with respect to infinite partial functions. In Section 3 we apply the idea of trace introduced in this proof to the still open question of whether analytic MAD families exist in $ {\omega}^{\omega}$. Using the idea of trace, we show that any analytic MAD families that may exist in $ {\omega}^{\omega}$ must satisfy strong combinatorial constraints. We also show that it is consistent to have MAD families in $ {\omega}^{\omega}$ that satisfy these constraints.


References [Enhancements On Off] (What's this?)

  • 1. Tomek Bartoszyński, Combinatorial aspects of measure and category, Fund. Math. 127 (1987), no. 3, 225-239. MR 917147 (88m:04001)
  • 2. Tomek Bartoszyński and Haim Judah, Set theory: On the structure of the real line, A K Peters Ltd., Wellesley, MA, 1995. MR 1350295 (96k:03002)
  • 3. James E. Baumgartner and Alan D. Taylor, Partition theorems and ultrafilters, Trans. Amer. Math. Soc. 241 (1978), 283-309. MR 0491193 (58:10458)
  • 4. Andreas Blass, Selective ultrafilters and homogeneity, Ann. Pure Appl. Logic 38 (1988), no. 3, 215-255. MR 942525 (89h:03081)
  • 5. Jörg Brendle, Otmar Spinas, and Yi Zhang, Uniformity of the meager ideal and maximal cofinitary groups, J. Algebra 232 (2000), no. 1, 209-225. MR 1783921 (2001i:03097)
  • 6. B. Kastermans, J. Steprāns, and Y. Zhang, Analytic and coanalytic families of almost disjoint functions, J. Symbolic Logic 73 (2008), no. 4, 1158-1172. MR 2467209
  • 7. Bart Kastermans, Very mad families, Advances in logic, Contemp. Math., vol. 425, Amer. Math. Soc., Providence, RI, 2007, pp. 105-112. MR 2322366 (2008c:03053)
  • 8. A. R. D. Mathias, Happy families, Ann. Math. Logic 12 (1977), no. 1, 59-111. MR 0491197 (58:10462)
  • 9. Arnold W. Miller, Arnie Miller's problem list, Set theory of the reals (Ramat Gan, 1991), Israel Math. Conf. Proc., vol. 6, Bar-Ilan Univ., Ramat Gan, 1993, pp. 645-654. MR 1234292 (94m:03073)
  • 10. Yi Zhang, On a class of m.a.d. families, J. Symbolic Logic 64 (1999), no. 2, 737-746. MR 1777782 (2001m:03104)
  • 11. -, Towards a problem of E. van Douwen and A. Miller, MLQ Math. Log. Q. 45 (1999), no. 2, 183-188. MR 1686175 (2000d:03113)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 03E17, 03E15, 03E05, 03E50

Retrieve articles in all journals with MSC (2010): 03E17, 03E15, 03E05, 03E50


Additional Information

Dilip Raghavan
Affiliation: Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 2E4
Email: raghavan@math.toronto.edu

DOI: https://doi.org/10.1090/S0002-9947-2010-04975-X
Keywords: Maximal almost disjoint family, cardinal invariants, analytic set
Received by editor(s): July 22, 2008
Published electronically: June 11, 2010
Additional Notes: The author was partially supported by NSF Grant DMS-0456653.
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society