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

Authors:
Menachem Kojman, Wiesław Kubiś and Saharon Shelah

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

Published electronically:
June 21, 2004

MathSciNet review:
2073313

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Abstract | References | Similar Articles | Additional Information

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

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