Barnabas Farkas

# Barnabas Farkas: Cardinal invariants versus towers in analytic P-ideals / An application of matrix iteration

06/11/17 21:20

Tuesday, November 7, 2017 17:15

Model1 of \(\mathrm{non}^*(\mathcal{I})=\mathfrak{c}\), there is a tower in \(\mathcal{I}\), and \(\mathrm{add}^*(\mathcal{I})<\mathrm{cov}^*(\mathcal{I})\). Method: Small filter iteration.

Model2 of \(\mathrm{non}^*(\mathcal{I})<\mathfrak{c}\), there is a tower in \(\mathcal{I}\), and \(\mathrm{add}^*(\mathcal{I})<\mathrm{cov}^*(\mathcal{I})\). Method: Matrix iteration.

This is a joint work with J. Brendle and J. Verner.

*Room:*D1-215*Barnabas Farkas (TU Wien)*

Speaker:Speaker:

*Title*: Cardinal invariants versus towers in analytic P-ideals / An application of matrix iteration*Abstract*. I will present two models concerning interactions between the existence of towers in analytic P-ideals and their cardinal invariants. It is trivial to see that if there is no tower in \(\mathcal{I}\), then \(\mathrm{add}^*(\mathcal{I})<\mathrm{cov}^*(\mathcal{I})\). I will prove that this implication cannot be reversed no matter the value of \(\mathrm{non}^*(\mathcal{I})\). More precisely, let \(\mathcal{I}\) be an arbitrary tall analytic P-ideal, I will construct the following two models:Model1 of \(\mathrm{non}^*(\mathcal{I})=\mathfrak{c}\), there is a tower in \(\mathcal{I}\), and \(\mathrm{add}^*(\mathcal{I})<\mathrm{cov}^*(\mathcal{I})\). Method: Small filter iteration.

Model2 of \(\mathrm{non}^*(\mathcal{I})<\mathfrak{c}\), there is a tower in \(\mathcal{I}\), and \(\mathrm{add}^*(\mathcal{I})<\mathrm{cov}^*(\mathcal{I})\). Method: Matrix iteration.

This is a joint work with J. Brendle and J. Verner.

# Barnabas Farkas: Towers in filters and related problems

26/04/16 14:02

Tuesday, May 10, 2016 17:15

*Room:*D1-215*Barnabas Farkas*

Speaker:Speaker:

*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 \(\geq\omega_2\).