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.