October 28
Corey Switzer,
University of Vienna
Ideal Independence, Filters and Maximal Sets of Reals
A family $\mathcal I \subseteq [\omega]^\omega$ is called ideal independent if given any finite, distinct $A, B_0, ..., B_{n-1} \in \mathcal I$, the set $A \setminus \bigcup_{i \lt n} B_i$ is infinite. In other words, the ideal generated by $\mathcal I \setminus \{A\}$ does not contain $A$ for any $A \in \mathcal I$. The least size of a maximal (with respect to inclusion) ideal independent family is denoted $\mathfrak{s}_{mm}$ and has recently been tied to several interesting questions in cardinal characteristics and Boolean algebra theory. In this talk we will sketch our new proof that this number is ZFC-provably greater than or equal to the ultrafilter number – the least size of a base for a non-principal ultrafilter on $\omega$. The proof is entirely combinatorial and relies only on a knowledge of ultrafilters and their properties. Time permitting, we will also discuss some interesting new applications of ideal independent families to topology via a generalization of Mrowka spaces usually studied for almost disjoint families. This is joint work with Serhii Bardyla, Jonathan Cancino and Vera Fischer.