List of problems on filters and ideals on $\omega$:

(permanently under construction, but even more so now...)

All filters considered are non-trivial and free and, dually, all ideals are non-trivial and contain all finite sets. Many of the problems are motivated by the study of the Katetov order defined as follows: Given ideals $\mathcal I$ and $\mathcal J$ on $\omega$ we say that $\mathcal I\leq_K\mathcal J$ if there is a function $f:\omega\to\omega$ such that $f^{-1}[I]\in\mathcal J$ for every $I\in\mathcal I$.

Borel ideals

1. Is there a Borel ideal Katetov minimal among tall Borel ideals?
2. Is $\mathcal R$ locally Katetov minimal among tall Borel ideals?
3. Does every tall Borel ideal contain a tall $F_\sigma$ ideal?

2. Is there a $\mathcal Z$-ultrafilter?
3. Is it consistent that $\mathcal I\leq_K \mathcal U^*$ for every Borel ideal $\mathcal I$ and every ultrafilter $\mathcal U$?