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On two problems of Erdos and Hechler: New methods in singular madness

Authors: Menachem Kojman, Wieslaw Kubis and Saharon Shelah
Journal: Proc. Amer. Math. Soc. 132 (2004), 3357-3365
MSC (2000): Primary 03E10, 03E04, 03E17, 03E35; Secondary 03E55, 03E50
Published electronically: June 21, 2004
MathSciNet review: 2073313
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Abstract: For an infinite cardinal $\mu$, $\operatorname{MAD}(\mu)$denotes the set of all cardinalities of nontrivial maximal almost disjoint families over $\mu$.

Erdos 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

Wieslaw Kubis
Affiliation: Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva, Israel – and – Institute of Mathematics, University of Silesia, Katowice, Poland

Saharon Shelah
Affiliation: Institute of Mathematics, Hebrew University of Jerusalem, Israel – and – Department of Mathematics, Rutgers University, New Brunswick, New Jersey

Keywords: Almost disjoint family, singular cardinal, bounding number, smooth pcf scales
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.
Article copyright: © Copyright 2004 American Mathematical Society