Tuesday, November 28, 2017 17:15

Room: D1-215

Speaker: Marcin Michalski

Title: Bernstein, Luzin and Sierpiński meet trees

Abstract. In [2] we have proven that if $\mathfrak{c}$ is a regular cardinal number, then the algebraic sum of a generalized Luzin set and a generalized Sierpiński set belongs to Marczewski ideal $s_0$. We will generalize this result for other tree ideals - $m_0$ and $l_0$ - using some lemmas on special kind of fusion sequences for trees of respective type.


Let us introduce a following notion. Let $\mathbb{X}$ be a set of trees.

Definition. We call a set $B$ a $\mathbb{X}$-Bernstein set, if for each $X\in \mathbb{X}$ we have $[X]\cap B\ne \mathrm{\varnothing }$.

We shall explore this notion for various set of trees, including Sacks, Miller and Laver trees, with the support of technics developed in [1].


[1] Brendle J., Strolling through paradise, Fundamenta Mathematicae, 148 (1995), pp. 1-25.

[2] Michalski M., Żeberski Sz., Some properties of I-Luzin, Topology and its Applications, 189 (2015), pp. 122-135.

Tags: Marcin Michalski

Tuesday, November 21, 2017 17:15

Room: D1-215

Speaker: Sakae Fuchino

Title: Downward Löwenheim-Skolem Theorems in stationary logic

Tags: Sakae Fuchino

Tuesday, November 14, 2017 17:15

Room: D1-215

Speaker: Tomasz Natkaniec

Title: Perfectly everywhere surjective but not Jones functions

Abstract. Given a function $f:\mathbb{R}\to \mathbb{R}$ we say that

1. $f$ is perfectly surjective ($f\in \mathrm{PES}$) if $f[P]=\mathbb{R}$ for every perfect set $P$;


2. $f$ is a Jones function ($f\in \mathrm{J}$) if $C\cap f\ne \mathrm{\varnothing }$ for every closed $C\subset {\mathbb{R}}^2$ with $\mathrm{dom}(C)$ of size $\mathfrak{c}$.
   

M. Fenoy-Munoz, J.L. Gamez-Merino, G.A. Munoz-Fernandez and E. Saez-Maestro in the paper A hierarchy in the family of real surjective functions [Open Math. 15 (2017), 486--501] asked about the lineability of the set $\mathrm{PES}\setminus \mathrm{J}$.
Answering this question we show that the class $\mathrm{PES}\setminus \mathrm{J}$ is ${\mathfrak{c}}^{+}$-lineable. Moreover, if
$2^{<\mathfrak{c}}=\mathfrak{c}$ then $\mathrm{PES}\setminus \mathrm{J}$ is $2^{\mathfrak{c}}$-lineable. We prove also that the additivity number
$A(\mathrm{PES}\setminus \mathrm{J})$ is between ${\omega }_1$ and $\mathfrak{c}$. Thus $A(\mathrm{PES}\setminus \mathrm{J})=\mathfrak{c}$ under CH,
however this equality can't be proved in ZFC, because the Covering Property Axiom CPA implies $A(\mathrm{PES}\setminus \mathrm{J})={\omega }_1<\mathfrak{c}$.

The talk is based on the joint paper:
K.C.Ciesielski, J.L. Gamez-Merino, T. Natkaniec, and J.B.Seoane-Sepulveda, On functions that are almost continuous and perfectly everywhere surjective but not Jones. Lineability and additivity, submitted.

Tags: Tomasz Natkaniec

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 ${\mathrm{add}}^{\ast }(\mathcal{I})<{\mathrm{cov}}^{\ast }(\mathcal{I})$. I will prove that this implication cannot be reversed no matter the value of ${\mathrm{non}}^{\ast }(\mathcal{I})$. More precisely, let $\mathcal{I}$ be an arbitrary tall analytic P-ideal, I will construct the following two models:

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

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

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

Tags: Barnabas Farkas