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Transactions of the American Mathematical Society
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
Published electronically: June 11, 2010
MathSciNet review: 2661500
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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.


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Additional Information

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

DOI: http://dx.doi.org/10.1090/S0002-9947-2010-04975-X
PII: S 0002-9947(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.