Wieslaw Kubis: Singular madness

Wednesday, March 26, 2014, 11:00
Prague – CTS

Speaker: Wieslaw Kubis (IM AS CR)

Title: Singular madness

Abstract:

Wieslaw Kubis will be presenting results from Kojman, Kubis, Shelah: On two problems of Erdős and Hechler: new methods in singular madness.

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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