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?**

**MAD families**

**1. (Steprans) Is there a Cohen-indestructible MAD family**

**2. Is there a Katetov maximal MAD family?**

**3. Is there a K-uniform MAD family?**

**Ultrafilters**

**1. Is there a Sacks-indestructible ultrafilter? **

**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 $\mathcal U$? **

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