November 2017

# Marcin Michalski: Bernstein, Luzin and Sierpiński meet trees

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 $$[X]\cap B\neq\emptyset$$.

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.

# Sakae Fuchino: Downward Löwenheim-Skolem Theorems in stationary logic

Tuesday, November 21, 2017 17:15

Room: D1-215

Speaker:
Sakae Fuchino

Title: Downward Löwenheim-Skolem Theorems in stationary logic

# Tomasz Natkaniec: Perfectly everywhere surjective but not Jones functions

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 $$C\cap f\neq\emptyset$$ 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.

# 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.