Barnabas Farkas

# Barnabas Farkas: A tool to avoid some technical forcing arguments when working with the Hechler forcing

17/05/23 14:02

Tuesday, May 23, 2023 17:00

*Location:*room**A.4.1 C-19***Barnabas Farkas (TU Wien)*

Speaker:Speaker:

*Title*: A tool to avoid some technical forcing arguments when working with the Hechler forcing*Abstract*: I'm going to present that virtually every result saying that finite support iterations of the Hechler forcing preserve a cardinal invariant being small and its dual being large can be reduced to a single preservation theorem. In other words, this theorem eliminates the technical forcing arguments from the proofs of these results and reduces them to easy coding exercises.# Barnabas Farkas: Degrees of destruction

23/02/19 07:54

Tuesday, February 26, 2019 17:15

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

Speaker:Speaker:

*Title*: Degrees of destruction*Abstract*. I'm going to present a survey on our results (joint with L. Zdomskyy) about the following strong notion of destroying Borel ideals: We say that the forcing notion \(\mathbb{P}\) \(+\)-destroys the Borel ideal \(\mathcal{I}\) if \(\mathbb{P}\) adds an \(\mathcal{I}\)-positive \(\dot{X}\) which has finite intersection with every \( A \in \mathcal{I}\cap V\). I will talk about the following:- Examples when usual destruction (that is, when \(\dot{X}\) required to be infinite only) implies \(+\)-destruction, and when it does not.
- Characterization of those Borel ideals which can be \(+\)-destroyed, in particular, we will see that if \(\mathcal{I}\) can be \(+\)-destroyed then the associated Mathias-Prikry forcing \(+\)-destroys it.
- Characterization of those analytic P-ideals which are \(+\)-destroyed by the associated Laver-Prikry forcing.

# 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 \(\geq\omega_2\).