Towers in filters, cardinal invariants, and Luzin type families

Abstract

We investigate which filters on $\omega$ can contain towers, that is, a modulo finite descending sequence without any pseudointersection (in $[\omega {]}^{\omega }$). We prove the following results: - Many classical examples of nice tall filters contain no towers (in ZFC). - It is consistent that tall analytic P-filters contain towers of arbitrary regular height (simultaneously for many regular cardinals as well). - It is consistent that all towers generate non-meager filters, in particular (consistently) Borel filters do not contain towers. - The statement "Every ultrafilter contains towers." is independent of ZFC. Furthermore, we study many possible logical (non)implications between the existence of towers in filters, inequalities between cardinal invariants of filters ($\mbox{add}^*(\mathcal F)$, $\mbox{cof}^*(\mathcal F)$, $\mbox{non}^*(\mathcal F)$, and $\mbox{cov}^*(\mathcal F)$), and the existence of Luzin type families (of size $\ge {\omega }_2$), that is, if $\mathcal{F}$ is a filter then $X\subseteq [\omega {]}^{\omega }$ is an $\mathcal{F}$-Luzin family if $\{A\in X:|A\setminus F|=\omega \}$ is countable for every $F\in \mathcal{F}$.