## 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