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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.]]>

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

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,

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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.]]>

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- If \(A\) is meagre (null) subset of real line, does there necessarily exist set \(B\) such that algebraic sum \(A+B\) doesn't have Baire property (is non-measurable)?
- If \(A\) is meagre (null) subset of real line, does there necessarily exist non-meagre (non-null) additive subgroup, disjoint with some translation of \(A\)?

It is not hard to prove that positive answer to 2. implies positive answer to 1, both for measure and category. We answer 2. affirmatively for category, while version for measure turns out to be independent of ZFC. The latter was essentially proved last year by A. Rosłanowski and S. Shelah. Both results holds for Cantor space with coordinatewise addition mod. 2 as well.]]>

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\(I_{\mathcal{F}} = \{A \subset \kappa \colon \bigcup_{\alpha \in A} F_\alpha \textrm{ is meager }, F_\alpha \in \mathcal{F}\}.\)

It would seem that the information about \(I_{\mathcal{F}}\) would give us full information about the ideal and the world in which it lives.

My talk is going to show that it is big simplification and localization technique from a Kuratowski partition cannot be omitted but the proof can be much simplier. During the talk I will show among others a new proof of non-existence of a Kuratowski partition in Ellentuck topology and a new combinatorial proof of Frankiewicz - Kunen Theorem (1987) on the existence of measurable cardinals. ]]>

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One of the open problems in the field of selection principles is to find the minimal hypothesis that the above properties can be separated in the class of sets of reals. Using purely

combinatorial approach, we provide examples under some set theoretic hypotheses. We apply obtained results to products of Menger spaces

This a joint work with Boaz Tsaban (Bar-Ilan University, Israel) and Lyubomyr Zdomskyy (Kurt Godel Research Center, Austria).

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\(X\) is \(1/n\)-homogeneous if \(X\) has exactly \(n\) orbits. In such a case we say that the degree of homogeneity of \(X\) equals \(n\). P. Pellicer Covarrubias, A. Santiago-Santos calculated the degree of homogeneity of connes over local dendrites depending on the degree of homogeneity of their bases. We will generalize above result on connes over locally connected curves.

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A structure is called ultrahomogeneous, if every embedding of its finitely generated substructure can be extended to an automorphism. Schmerl showed that there are only countably many, up to isomorphism, ultrahomogeneous countable partially ordered sets. More precisely he proved the following characterization:

Let \((H, <)\) be a countable partially ordered set. Then \((H, <)\) is ultrahomogeneous iff it is isomorphic to one of the following:

- \((A_n, <)\) for \(1 \le n \le\omega\);
- \((B_n, <)\) for \(1 \le n \le\omega\);
- \((C_n, <)\) for \(1 \le n \le\omega\);
- \((D, <)\).

Moreover, no two of the partially ordered sets listed above are isomorphic. Consider automorphisms groups \(Aut(A_\omega) = S_\infty\), \(Aut(B_n) \), \(Aut(C_n)\) and \(Aut(D)\). We prove that each of these groups contains two elements f, g such that the subgroup generated by f and g is free and dense. By Schmerl’s Theorem the automorphism group of a countable infinite partially ordered set is freely topologically 2-generated.

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Our remainder is a Stone space of a Boolean subalgebra of Lebesgue measurable subsets of \(2^{\omega}\) containing all clopen sets.]]>

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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.]]>

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One of the major open problems in the field of selection principles is whether there are, in ZFC, two Menger sets of real numbers whose product is not Menger. We provide examples under various set theoretic hypotheses, some being weak portions of the Continuum Hypothesis, and some violating it. The proof method is new.]]>

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We will show that if \(2^\omega\) is a regular cardinal then for every generalized Luzin set \(L\) and every generalized Sierpiński set \(S\) an algebraic sum \(L+S\) belongs to the Marczewski ideal \(s_0\) (i.e. for every perfect set \(P\) there exists a perfect set \(Q\) such that \(Q\subseteq P\) and \(Q\cap (L+S)=\emptyset\)). To prove the theorem we shall prove and use a generalized version of the Rothberger theorem.

We will also formulate a series of results involving algebraic, topological and measure structure of the real line, that emerged during searching for a proof of the above theorem.]]>

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\[\rho(f,g)=\min\{1,\sup\limits_{x\in\mathbb{R}}|f(x)-g(x)|\}\quad \text{for \(f,g\in\mathbb{R}^\mathbb{R}\)}\]

and the following subsets of \(\mathbb{R}^\mathbb{R}\):

- Darboux functions (\(f\in\mathscr{D}\) if whenever \(a < b\) and \(y\) is a number between \(f(a)\) and \(f(b)\), there exists an \(x_0\in(a, b)\) such that \(f(x_0) = y\)).
- quasi-continuous functions (\(f\in\mathscr{Q}\) if it is quasi-continuous at any point \(x\in\mathbb{R}\)).

Function \(f\) is*quasi-continuous*at \(x \in \mathbb{R}\) if for any open interval \(I\ni x\) and each \(\varepsilon>0\) there exists a nontrivial interval \(J\subset I\) such that \({\rm diam} (f[J\cup \{x\}]) <\varepsilon\). - Świątkowski functions ( \(f \in \mathscr{\acute S}\) if for all \( a < b \) with \(f(a) \ne f(b)\), there is a \(y\) between \(f(a)\) and \(f(b)\) and an \(x\in(a,b) \cap \mathcal{C}(f)\) such that \(f(x)=y\), where \(\mathcal{C}(f)\) denotes the set of all continuity points of function \(f\)).
- Świątkowski functions (\(f\in\mathscr{\acute S}_s\) if for all \(a < b\) and each \(y\) between \(f(a)\) and \(f(b)\) there is an \(x\in(a,b) \cap \mathcal{C}(f)\) such that \(f(x)=y\)).

The aim of this is to compare this sets in terms of porosity.

Let \((X,d)\) be a metric space, \(x\in A\subset M\), and \(r\in\mathbb{R}_+\). We define

\[\gamma(x,r,M)=\sup\{{t\geq 0}:\ \exists_{z\in M} B(z,t)\subset B(x,r)\setminus M\}\]

and

\[p^u(M, x)=2\limsup\limits_{t\to r^+}\frac{\gamma(x,r,M)}{r}.\]

\[p_l(M, x)=2\liminf\limits_{t\to r^+}\frac{\gamma(x,r,M)}{r}.\]

Quantity \(p^u(M,x)\) is called upper porosity of \(M\) at the point \(x\). We say that \(M\) is upper \(p-\)porous if \(p=\inf\{p^u(M,x):\ x\in M\}>0\).

Analogously we define lower porosity.]]>

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Moreover, I will discuss the existence of large midpoint-free subsets of arbitrary subset of the real line.]]>

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My recent study concerns the problem how (and whether at all) we can describe the structure of the class of Boolean algebras with the Nikodym property in terms of well-known objects occuring inside \(\wp(\omega)\) or \(\omega^\omega\), e.g. countable Boolean algebras, dominating families, Lebesgue null sets etc. During my talk I will present an attempt to obtain such a description via families of antichains in countable subalgebras of \(\wp(\omega)\) having some special measure-theoretic properties.]]>

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I will discuss some properties of ideals obtained this way (among others, I will show that they can be generated using Solecki's submeasures). I will then examine inclusions between ideals obtained for different functions \(g\).

I will also discuss connections between our ideals, "density-like" ideals and Erdos-Ulam ideals. I will present joint results with M. Balcerzak, P. Das and M. Filipczak.]]>

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We will give generalizations of given notion to the case of arbitrary metric space. We will analyze algebraic and set-theoretic properties of the family of microscopic sets.]]>

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Let \(\kappa\) be a cardinal number and let \(\mathcal{L}\) be a commutative algebra. Assume that \(A\subseteq\mathcal{L}\). We say that \(A\) is:

- \(\kappa\)-
*algegrable*if \(A\cup \{0\}\) contains \(\kappa\)-generated algebra \(B\); *strongly*\(\kappa\)-*algegrable*if \(A\cup \{0\}\) contains \(\kappa\)-generated free algebra \(B\).

In many recent articles authors studied algebrability of sets naturally appering in mathematical analysis. It seems that required results are the general methods of algebrability which can cover known methods and give new constructions.

We will describe two methods:

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