Robert Rałowski

# Robert Rałowski: Bernstein set and continuous functions

25/02/16 15:41

Tuesday, March 1, 2016 17:15

We show some consistency results for completely nonmeasurable sets with respect to \(\sigma\)-ideals of null sets and meager sets on the real line.

These results was obtained commonly with Jacek Cichoń, Michał Morayne and me.

*Room:*D1-215*Robert Rałowski*

Speaker:Speaker:

*Title*: Bernstein set and continuous functions*Abstract*. Alexander V. Osipov asked "It is true that for any Bernstein subset \(B\subset \mathbb{R}\) there are countable many continous functions from \(B\) to \(\mathbb{R}\) such that the union of images of \(B\) is a whole real line \(\mathbb{R}\)". We give the positive answer for this question, but we show that this result is not true for a \(T_2\) class of functions.We show some consistency results for completely nonmeasurable sets with respect to \(\sigma\)-ideals of null sets and meager sets on the real line.

These results was obtained commonly with Jacek Cichoń, Michał Morayne and me.

# Robert Rałowski: Two point sets, continuation

23/03/15 20:46

Tuesday, March 24, 2015 17:15

*Room:*D1-215*Robert Rałowski*

Speaker:Speaker:

*Title*: Two point sets, continuation*Abstract*. We will continue discussion started a week ago concerning two point sets. We will give another example of a property of two point set which is consistent with ZFC.# Robert Rałowski: Cohen indestructible mad families in partial two point sets

11/03/15 17:35

Tuesday, March 17, 2015 17:15

*Room:*D1-215*Robert Rałowski*

Speaker:Speaker:

*Title*: Cohen indestructible mad families in partial two point sets*Abstract*. We discuss on classical construction of Cohen indestructible mad family given by Kenneth Kunen and we apply this method to obtain a partial Cohen indestructible mad family in Baire space as a canonical copy of the real plane.# Robert Rałowski: On generalized Luzin sets

05/12/14 18:34

Tuesday, December 9, 2014 17:15

*Room:*D1-215*Robert Rałowski*

Speaker:Speaker:

*Title*: On generalized Luzin sets*Abstract*. We will show results obtained together with Sz. Żeberski concerning properties of \((I,J)\)-Luzin sets (for \(I, J\) \(\sigma\)-ideals on Polish space). Under some settheoretical assumptions we will construct \(\mathfrak{c}\) many generalized Luzin sets which are not Borel equivalent. We will also examine some forcing notions which do not kill generalized Luzin sets.# Robert Rałowski: On m.a.d. \(s_0\)-nonmeasurable sets with a small dominating subfamilies

27/10/14 16:02

Tuesday, October 28, 2014 17:15

*Room:*D1-215*Robert Rałowski*

Speaker:Speaker:

*Title*: On m.a.d. \(s_0\)-nonmeasurable sets with a small dominating subfamilies*Abstract*. We show that \(\mathfrak{d}=\aleph_1\) implies the existence of maximal familiy of eventually different reals on Baire space which forms a nonmeasurable set with respect to an ideals generated by trees (perfect, Laver or Miller trees for example).# Robert Rałowski: Nonmeasurability with respect to Marczewski ideal

09/10/14 17:51

Tuesday, October 14, 2014 17:15

*Room:*D1-215*Robert Rałowski*

Speaker:Speaker:

*Title:*Nonmeasurability with respect to Marczewski ideal*Abstract:*Among the others we show relative consistency of ZFC theory with \(\aleph_1< 2^{\aleph_0}\) and there is a nonmesurable (with respect to ideal generated by complete Laver trees) m.a.d. family \(\mathcal{A}\) on Baire space \(\omega^\omega\). In ZFC there is a subset \(\mathcal{A}’\subseteq \mathcal{A}\) of size \(\aleph_1\) unbounded in \(\omega^\omega\). We show that there is m.a.d. family which is nonmeasurable with respect to Marczewski ideal.