Barnabas Farkas: Towers in filters and related problems

Tuesday, May 10, 2016, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Barnabas Farkas (University of Vienna)

Title: Towers in filters and related problems

Abstract:

I am going to present a survey on my recently finished joint work with J. Brendle and J. Verner. In this paper we investigated which filters can contain towers, that is, a $\subseteq^*$-decreasing sequence in the filter without any pseudointersection (in $[\omega]^\omega$). I will present Borel examples which contain no towers in $\mathrm{ZFC}$, and also examples for which it is independent of $\mathrm{ZFC}$. I will prove that consistently every tower generates a non-meager filter, in particular (consistently) Borel filters cannot contain towers. And finally, I will present the “map” of logical implications and non-implications between (a) the existence of a tower in a filter $\mathcal{F}$, (b) inequalities between cardinal invariants of $\mathcal{F}$, and (c) the existence of a peculiar object, an $\mathcal{F}$-Luzin set of size $\geq\omega_2$.

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