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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On two problems of Erdos and Hechler: New methods in singular madness
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by Menachem Kojman, Wiesław Kubiś and Saharon Shelah PDF
Proc. Amer. Math. Soc. 132 (2004), 3357-3365 Request permission

Abstract:

For an infinite cardinal $\mu$, $\operatorname {MAD}(\mu )$ denotes the set of all cardinalities of nontrivial maximal almost disjoint families over $\mu$. Erdős and Hechler proved in 1973 the consistency of $\mu \in \operatorname {MAD}(\mu )$ for a singular cardinal $\mu$ and asked if it was ever possible for a singular $\mu$ that $\mu \notin \operatorname {MAD}(\mu )$, and also whether $2^{\operatorname {cf}\mu } <\mu \Longrightarrow \mu \in \operatorname {MAD}(\mu )$ for every singular cardinal $\mu$. We introduce a new method for controlling $\operatorname {MAD} (\mu )$ for a singular $\mu$ and, among other new results about the structure of $\operatorname {MAD}(\mu )$ for singular $\mu$, settle both problems affirmatively.
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Additional Information
  • Menachem Kojman
  • Affiliation: Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva, Israel
  • Email: kojman@math.bgu.ac.il
  • Wiesław Kubiś
  • Affiliation: Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva, Israel – and – Institute of Mathematics, University of Silesia, Katowice, Poland
  • Email: kubis@math.bgu.ac.il
  • Saharon Shelah
  • Affiliation: Institute of Mathematics, Hebrew University of Jerusalem, Israel – and – Department of Mathematics, Rutgers University, New Brunswick, New Jersey
  • MR Author ID: 160185
  • ORCID: 0000-0003-0462-3152
  • Email: shelah@math.huji.ac.il
  • Received by editor(s): June 10, 2002
  • Received by editor(s) in revised form: September 10, 2002
  • Published electronically: June 21, 2004
  • Additional Notes: The first author’s research partially supported by an Israeli Science Foundation grant no. 177/01
    The third author’s research was supported by The Israel Science Foundation, Publication 793.
  • Communicated by: Carl G. Jockusch, Jr.
  • © Copyright 2004 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 132 (2004), 3357-3365
  • MSC (2000): Primary 03E10, 03E04, 03E17, 03E35; Secondary 03E55, 03E50
  • DOI: https://doi.org/10.1090/S0002-9939-04-07580-X
  • MathSciNet review: 2073313