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

Kojman, Menachem; Kubiś, Wiesław; Shelah, Saharon

doi:10.1090/S0002-9939-04-07580-X

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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