Barnabas Farkas

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

Tuesday, November 7, 2017 17:15

Room: D1-215

Speaker:
Barnabas Farkas (TU Wien)

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

Tuesday, May 10, 2016 17:15

Room: D1-215

Speaker:
Barnabas Farkas

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 $$\geq\omega_2$$.