Szymon Żeberski

# Szymon Żeberski: Remarks on Eggleston theorem

21/03/22 14:54

Tuesday, March 22, 2022 17:00

*Location:*room 605, Mathematical Institute, University of Wroclaw*Szymon Żeberski*

Speaker:Speaker:

*Title*: Remarks on Eggleston theorem*Abstract*: We will discuss possible variants and generalizations of Eggleston theorem about inscribing big rectangles into big subsets of the plane. We will focus mainly on product of two Cantor spaces and comeager sets.# Szymon Żeberski: Applications of non-measurable unions

27/04/21 04:53

Tuesday, April 27, 2021 17:00

It is a joint work with Taras Banakh and Robert Rałowski https://arxiv.org/abs/2011.11342.

*Location:***Zoom.us**: if you want to participate please contact organizers*Szymon Żeberski*

Speaker:Speaker:

*Title*: Applications of non-measurable unions*Abstract*: Using a game-theoretic approach (Set-Cover game) we obtain a generalization of the classical result of Brzuchowski, Cichoń, Grzegorek and Ryll-Nardzewski on non-measurable unions. We will present applications of this result to establishing some countability and continuity properties of measurable functions and homomorphisms between topological groups.It is a joint work with Taras Banakh and Robert Rałowski https://arxiv.org/abs/2011.11342.

# Szymon Żeberski: Mycielski theorem and Miller trees

04/04/19 15:37

Tuesday, April 9, 2019 17:15

We will discuss how far this can be generalized if we replace perfect set by superperfect set, i.e a body of a Miller tree.

It turns out that there is a comeager \(A\subseteq (\omega^\omega)^2\) such that \(A\cup \Delta\) does not contain any set of the form \(M\times M\), where \(M\) is superperfect.

However, for comeager \(A\subseteq [0,1]^2\) one can find a perfect set \(P\) and a superperfect set \(M\supseteq P\) such that \(P\times M\subseteq A\cup\Delta\).

We will also discuss measure case, where results are slightly different.

*Room:*D1-215*Szymon Żeberski*

Speaker:Speaker:

*Title*: Mycielski theorem and Miller trees*Abstract*. The classical Mycielski theorem says that for comeager \(A\subseteq [0,1]^2\) one can find a perfect set \(P\) such that \(P\times P\subseteq A\cup\Delta\). (The same is true if we start with \(A\) of measure 1.)We will discuss how far this can be generalized if we replace perfect set by superperfect set, i.e a body of a Miller tree.

It turns out that there is a comeager \(A\subseteq (\omega^\omega)^2\) such that \(A\cup \Delta\) does not contain any set of the form \(M\times M\), where \(M\) is superperfect.

However, for comeager \(A\subseteq [0,1]^2\) one can find a perfect set \(P\) and a superperfect set \(M\supseteq P\) such that \(P\times M\subseteq A\cup\Delta\).

We will also discuss measure case, where results are slightly different.

# Szymon Żeberski: Applications of Shoenfield Absoluteness Lemma

27/05/15 21:38

Tuesday, June 2, 2015 17:15

*Room:*D1-215*Szymon Żeberski*

Speaker:Speaker:

*Title*: Applications of Shoenfield Absoluteness Lemma*Abstract*. We will recall Shoenfield Absoluteness Lemma about \(\Sigma^1_2\) sentences. We will show applications of this theorem connected to topological and algebraic structure of Polish spaces in publications co-authored by the speaker.# Szymon Żeberski: An example of a capacity for which all positive Borel sets are thick

27/02/15 16:07

Tuesday, March 3, 2015 17:15

*Room:*D1-215*Szymon Żeberski*

Speaker:Speaker:

*Title*: An example of a capacity for which all positive Borel sets are thick*Abstract*. The result is obtained together with Michał Morayne. We will show an example of a capacity on Cantors cube for which all positive Borel sets can be partitioned into continuum many positive Borel sets.# Szymon Żeberski: \(\sigma\)-ideals invariant under measure-preserving homeomorphisms on Cantor's cube

02/10/14 21:07

Tuesday, October 7, 2014 17:15

*Room:*D1-215*Szymon Żeberski*

Speaker:Speaker:

*Title:*\(\sigma\)-ideals invariant under measure-preserving homeomorphisms on Cantor's cube*Abstract:*Results were obtained together with Taras Banakh and Robert Rałowski. We will show that there are only four nontrivial sigma-ideals with Borel base invariant under measure preserving homeomorphisms on Cantor's cube. These ideals are: \(\mathscr{E}\), \(\mathscr{M} \cap \mathscr{N}\), \(\mathscr{M}\), \(\mathscr{N}\).