36th Summer Topology Conference

"Ultrafilters and Ideal Independent Families"

Switzer, Corey

A family

$\mathcal I \subseteq [\omega]^\omega$ is called ideal independent if for every

$A \in \mathcal I$ and every finite

$F \subseteq \mathcal I \setminus \{A\}$ we have

$A \nsubseteq^* \bigcup F$ . In other words

$A$ is not in the ideal generated by

$\mathcal I \setminus \{A\}$ . The cardinal

$\mathfrak{s}_{mm}$ is defined as the minimal size a of a maximal ideal independent family. In this talk we will discuss how this cardinal relates to other cardinal characteristics of extremal sets of reals. In particular we will show that

$\mathfrak{s}_{mm}$ is independent the independence number

$\mathfrak{i}$ , but surprisingly,

$\mathsf{ZFC}$ -provably greater than or equal to the ultrafilter number

$\mathfrak{u}$ . This is joint work with Jonathan Cancino and Vera Fischer.

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