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This is a joint work with Miguel A. Cardona and Diego A. Mejía.

References:

[CMU] Miguel A. Cardona, Diego A. Mejía and Andrés F. Uribe-Zapata. Finitely additive measures on Boolean algebras. In Preparation.

[UZ23] Andrés Uribe-Zapata. Iterated forcing with finitely additive measures: applications of probability to forcing theory. Master’s thesis, Universidad Nacional de Colombia, sede Medellín, 2023. https://shorturl.at/sHY59.]]>

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In this talk we will study the family \(\mathcal{AN}\) of such ideals \(\mathcal{I}\) on \(\omega\) that the space \(N_{\mathcal{I}^*}\) carries a sequence \(\langle\mu_n\colon n\in\omega\rangle\) of finitely supported signed measures satisfying \(\|\mu_n\|\rightarrow\infty\) and \(\mu_n(A)\rightarrow 0\) for every \(A\in Clopen(N_{\mathcal{I}^*})\). If \(\mathcal{I}\in\mathcal{AN}\) and \(N_{\mathcal{I}^*}\) is embeddable into the Stone space \(St(\mathcal{A})\) of a given Boolean algebra \(\mathcal{A}\), then \(\mathcal{A}\) does not have the Nikodym property.

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Every \(\omega\)-Corson compact space is Eberlein compact. For a Tichonoff space \(X\), let \(C_p(X)\) denote the space of real continuous functions on \(X\) endowed with the pointwise convergence topology.

During the talk we will show that the class \(\omega\)-Corson compact spaces \(K\) is invariant under linear homeomorphism of function spaces \(C_p(K)\) and other related results.]]>

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In this talk we will study the family \(\mathcal{AN}\) of such ideals \(\mathcal{I}\) on \(\omega\) that the space \(N_{\mathcal{I}^*}\) carries a sequence \(\langle\mu_n\colon n\in\omega\rangle\) of finitely supported signed measures satisfying \(\|\mu_n\|\rightarrow\infty\) and \(\mu_n(A)\rightarrow 0\) for every \(A\in Clopen(N_{\mathcal{I}^*})\). If \(\mathcal{I}\in\mathcal{AN}\) and \(N_{\mathcal{I}^*}\) is embeddable into the Stone space \(St(\mathcal{A})\) of a given Boolean algebra \(\mathcal{A}\), then \(\mathcal{A}\) does not have the Nikodym property.]]>

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This is a joint work with Tomasz Weiss and Lyubomyr Zdomskyy.

The research was funded by the National Science Centre, Poland and the Austrian Science Found under the Weave-UNISONO call in the Weave programme, project: Set-theoretic aspects of topological selections 2021/03/Y/ST1/00122]]>

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It was proven by Fremlin that, under Martin's axiom and negation of continuum hypothesis, for a compact Hausdorff space \(K\) the existance of a Radon probability of uncountable type is equivalent to the exitence of a continuous surjection from \(K\) onto \([0,1]^{\omega_1}\). Hence, under such assumptions, countable tightness of \(P(K)\) implies that there is no Radon probability on \(K\) which has uncountable type. Later, Plebanek and Sobota showed that, without any additional set-theoretic assumptions, countable tightness of \(P(K\times K)\) implies that there is no Radon probability on \(K\) which has uncountable type as well. It is thus natural to ask whether the implication "\(P(K)\) has countable tightness implies every Radon probability on \(K\) has countable type" holds in ZFC.

I will present our joint result with Piotr Koszmider that under diamond principle there is a compact Hausdorff space \(K\) such that \(P(K)\) has countable tightness but there exists a Radon probability on \(K\) of uncountable type.

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A compact space \(K\) is \(\omega\)-Corson compact if, for some set \(\Gamma\), \(K\) is homeomorphic to a subset of the \(\sigma\)-product of real lines \(\sigma(\mathbb{R}^\Gamma)\), i.e. the subspace of the product \(\mathbb{R}^\Gamma\) consisting of functions with finite supports. Clearly, every \(\omega\)-Corson compact space is Eberlein compact.

We will present a characterization of \(\omega\)-Corson compact spaces, and some other results concerning this class of spaces and related classes of Eberlein compacta.

This is a joint research with Grzegorz Plebanek and Krzysztof Zakrzewski, see

https://arxiv.org/abs/2107.02513

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This is a joint work with Valentin Haberl and Lyubomyr Zdomskyy.

The research was funded by the National Science Centre, Poland and the Austrian Science Found under the Weave-UNISONO call in the Weave programme, project: Set-theoretic aspects of topological selections 2021/03/Y/ST1/00122.]]>

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We show that there exists a sequence of elements of \(\mathfrak M\) such that their mutual distances are > 1/2. It seems to be an open problem whether "1/2" can be replaced here by a bigger constant C. We show that C must be smaller than 9/14. Moreover, we present a version of the problem in terms of binary codes.]]>

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The talk will be devoted to some results obtained in a recent joint work with Paul Szeptycki (Canada). The aim of this work is to better understand the boundaries of the class $\Delta$, by presenting new examples and counter-examples.

1) We examine when trees considered as topological spaces equipped with the interval topology belong to \(\Delta\).

In particular, we prove that no Souslin tree is a \(\Delta\)-space. Other main results are connected with the study of

2) \(\Psi\)-spaces built on maximal almost disjoint families of countable sets; and

3) Ladder system spaces.

There exists an Isbell-Mrówka \(\Psi\)-space \(X\) (which is in \(\Delta\)) such that one-point extension \(X_p = X \cup \{p\}\) of \(X\) has uncountable tightness at the point \(p\), for some \(p \in \beta(X) \setminus X\).

It is consistent with CH that all ladder system spaces on \(\omega_1\) are \(\Delta\)-spaces.

We show that in forcing extension of ZFC obtained by adding one Cohen real, there is a ladder system space on \(\omega_1\) which is not in \(\Delta\).

[1] Jerzy Kąkol and Arkady Leiderman, A characterization of \(X\) for which spaces \(C_p(X)\) are distinguished and its applications, Proc. Amer. Math. Soc., series B, 8 (2021), 86-99.

[2] Jerzy Kąkol and Arkady Leiderman, Basic properties of \(X\) for which the space \(C_p(X)\) is distinguished, Proc. Amer. Math. Soc., series B, (8) (2021), 267-280.

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The presented results are from the paper “Katětov order on Borel ideals” by Michael Hrusak.]]>

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[1] J. Bobok, P. Pyrih and B. Vejnar, Non-cut, shore and non-block points in continua, Glas. Mat. Ser. III 51 (71) (2016), 237–253.]]>

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It is not hard to see that (a characteristic function of) a P-point is a P-measure. However, a question whether the existence of P-measures implies the existence of P-points remains open.

I will talk about current knowledge of the problem including my and Piotr Borodulin-Nadzieja's efforts and results - based on the Silver forcing and its variations.]]>

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Joint work with Krzysztof Omiljanowski.]]>

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- \((S^+)\): \((A+B)\neq\emptyset\) for every \(A,B\in\mathcal{B}(X)\setminus \mathcal{F}\),
- \((S^-)\): \(0\in (A-A)\) for every \(A\in\mathcal{B}(X)\setminus \mathcal{F}\),
- \((D^+)\): \(A+B\) is non-meager for every \(A,B\in\mathcal{B}(X)\setminus \mathcal{F}\),
- \((D^-)\): \(A-A\) is non-meager for every \(A\in\mathcal{B}(X)\setminus \mathcal{F}\).

It is known that the family \(\mathcal{M}\) of all meager sets as well as the family \(\mathcal{N}\) of all sets of Haar measure zero satisfy each of these conditions. We prove that the family \(\mathcal{M}\cap\mathcal{N}\) satisfies \((S^-)\), \((D^+)\), \((D^-)\) although it does not satisfy \((S^+)\). We also show that the \(\sigma\)-ideal \(\sigma\overline{\mathcal{N}}\subset \mathcal{M}\cap\mathcal{N}\) generated by closed sets of Haar measure zero satisfies only \((D^+)\) and \((D^-)\) which leads us to the Laczkovich property. This is joint work with T. Banakh, I. Banakh, Sz. Głąb and J. Swaczyna.

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The question arises, whether \(A^{\flat}\) is an ordinary attractor. The answer is `yes'.

\begin{eqnarray*}

\mathcal{G}

= (X; \;\; g^{-j}\circ f_i\circ g^j \;\;: i\in I, j\in\mathbb{Z}_p),

%\label{eq:IFS-Gconj}

\\

GF

= (X; \;\; g^k\circ f_i\circ g^j \;\;: i\in I, j,k\in\mathbb{Z}_p),

%\label{eq:IFS-GF}

\\

\widehat{\mathcal{G}} =

(X; \;\; g^k\circ f_i \;\;: i\in I, k\in\mathbb{Z}_p).

%\label{eq:IFS-Gdoubletilde}

\end{eqnarray*}

Moreover, the attractor \(A_{\widetilde{\mathcal{G}}}\) of a contractive IFS \(\widetilde{\mathcal{G}} = (X; f_i\circ g^j: i\in I, j\in\mathbb{Z}_p)\) is a smaller copy of \(A^{\flat}\): \(A_{\mathcal{F}} \subset A_{\widetilde{\mathcal{G}}} \subset A^{\flat} = \bigcup_{k=0}^{p-1} g^k(A_{\widetilde{\mathcal{G}}})\).

The question whether the disjunctive chaos game algorithm is valid for the enriched IFS \(\widetilde{\mathcal{F}}\) leads to interesting problems in combinatorics on words. Finally, to allow for similar results in case \(g\) is a non-periodic isometry, or \(\mathcal{F}\) is enriched by more than one isometry, one needs to employ infinite IFSs (F. Strobin, 2021).]]>

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The presented results are from an upcoming paper with O. Guzman and M. Hrusak.]]>

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We explain the aim of the programme and then discuss a joint work with Wiesław Kubiś on a specific way of constructing structures of size \(\aleph_1\) using finite approximations, namely by organising the approximations along a simplified morass. We demonstrate a connection with Fraïssé limits and show that the naturally obtained structure of size \(\aleph_1\) is homogeneous. We give some examples of interesting structures constructed, such as a homogeneous antimetric space of size \(\aleph_1\). Finally, we comment on the situation when one Cohen real is added.]]>

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For a theory \(T\) , \(T_{\exists\lor\forall}\) denotes the logical consequences of \(T\) which are boolean combinations of universal sentences. \(T\) is the AMC of \(T^*\) if it is model complete and \(T_{\exists\lor\forall}=T^*_{\exists\lor\forall}\).

The \(\{+, ·, 0, 1\}\)-theory ACF of algebraically closed field is the model companion of the theory of Fields but not its AMC as \(\exists x (x^2+1=0)\in ACF_{\exists\lor\forall}\setminus Fields_{\exists\lor\forall}\).

We use AMC to study the continuum problem and to gauge the expressive power of forcing. We show that (a definable version of) \(2^{\aleph_0} = \aleph_2\) is the unique solution to the continuum problem which can be in the AMC of a partial Morleyization of the \(\in\)-theory

ZFC+there are class many supercompact cardinals. We also show that (assuming large cardinals) forcibility overlaps with the apparently stronger notion of consistency for any mathematical problem \(\varphi\) expressible as a \(\Pi_2\) -sentence of a (very large fragment of) third order arithmetic (CH, the Suslin hypothesis, the Whitehead conjecture for free groups, are a small sample of such problems \(\varphi\)).

Partial Morleyizations can be described as follows: let \(Form_{\tau}\) be the set of first order \(\tau\)-formulas; for a subset A of \(Form_{\tau}\), \(\tau_A\) is the expansion of \(\tau\) adding atomic relation symbols \(R_\varphi\) for all formulas \(\varphi\in A\) and \(T_{\tau,A}\) is the \(\tau_A\)-theory asserting that each \(\tau\)-formula \(\varphi(x)\in A\) is logically equivalent to the corresponding atomic formula \(R_\varphi (x\sim x)\). For a \(\tau\)-theory T, \(T + Ti_{\tau,A}\) is the partial Morleyization of T induced by \(A\subseteq F_\tau\).]]>

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

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This is joint work with Jeffrey Bergfalk and Martino Lupini.]]>

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The talk is based on the paper "On two problems concerning Eberlein compacta": http://arxiv.org/abs/2103.03153]]>

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This is a joint work (in progress) with Antonio Avilés and Abraham Rueda Zoca.]]>

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Some lemmas similar to that in the paper of Bucchioni were used earlier to prove the equivalence of the convergence in category and the Cauchy condition for this type of convergence.]]>

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Sealing is a generic absoluteness statement which was introduced by Woodin. First given a generic object \(g\), let \(\Gamma^\infty_g\) be the set of universally Baire sets of \(V[g]\) and \(R_g\) be the set of reals of \(V[g]\).

Sealing (essentially) says that for all \(V\)-generic \(g\) and all \(V[g]\)-generic \(h\) there is an embedding

\(j: L(\Gamma^\infty_g, R_g)\to L(\Gamma^\infty_g*h, R_g*h).\)

Thus, in a way, Sealing says that there cannot be independence results about universally Baire sets, and as such it is a generalization of Shoenfield's absoluteness theorem.

It is an open problem if large cardinals imply Sealing. No canonical inner model can satisfy it, and so if some large cardinal implies it then its inner model theory must be significantly different than the current theory we have. Surprisingly, Woodin showed that if there are proper class of Woodin cardinals and delta is a supercompact then collapsing \(2^{2^\delta}\) to be countable forces Sealing. Because of its impact on the inner model problem and because of Woodin's result, it seemed that the set theoretic strength of Sealing must be at the level of supercompact cardinals. However, the speaker and Nam Trang showed that it is weaker than a Woodin cardinal that is a limit of Woodin cardinals (which are significantly smaller than supercompact cardinals). We will exposit this theorem and will also explain its consequences on the inner model problem. ]]>

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In our upcoming JSL paper with Saharon Shelah we prove that this is not the case: under MA(\(\omega_1\)) there is no universal wide Aronszajn tree.

The talk will discuss that paper. The paper is available on the arxiv and on line at JSL in the preproof version DOI: 10.1017/jsl.2020.42 ]]>

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1) \((X,\rho)\) is separable metric space,

2) identity \(id:(X,\rho) \to X\) is continuous,

2) every \(\rho\)-Cauchy sequence is converged in \(X.\)

Family \(\mathcal A \subseteq P(X)\) is analytic if

1) \(X\in \mathcal A\)

2) \(\mathcal A\) is closed on intersections

3) each \(A\in{\mathcal A}\) has analytic metric \(\rho\) and for any \(\epsilon>0\) there is a countable cover \(\mathcal U \subseteq \mathcal A\) of \(\mathcal A\) with \(\epsilon\) \(\rho\)-diameter.

We present a theorem that generalizes the well known result obtained by Brzuchowski, Cichoń, Grzegorek and Ryll-Nardzewski about nonmeasurable unions.

Theorem. Let \({\mathcal A}\) be an analytic family of Hausdorff space \(X\), any \(I\) \(\sigma\)-ideal of \(X.\) If \( J\subseteq I\) is point-finite family such that \(\bigcup J \notin I\) then there is a subfamily \(J' \subseteq J\) and \(A\in {\mathcal A}\) such that

1) \(A\cap \bigcap J' \notin I\)

2) for every \(A' \in {\mathcal A}\) if \(A' \subseteq A\cap \bigcup J'\) then \(A' \in I.\)

We show that the above Theorem implies the Theorem on nomeasurabie unions with respect to tree ideals like Marczeski ideal \(s_0\) for example.

Moreover, the above Theorem implies theorem on nonmeasurable unions with respect to \(\sigma\)-ideals which has Marczewski-Burstin representation.

The last mentioned result gives a theorem about nonmeasurable unions with respect to the ideal of Ramsey-null set in Ramsey space with Ellentuck topology.

The talk is based on a joint work with Taras Banakh and Szymon Żeberski.]]>

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The continuum (\(=2^{\aleph_0}\)) is either 1. \(\aleph_1\) or 2. \(\aleph_2\) or 3. fairly large.

Here, the fair largeness of the continuum can be expressed either in terms of weak mahloness and/or some other ``large'' cardinal notions compatible with the continuum, or even in terms of existence of some saturated ideals.

The reflection principles we consider here can be formulated as the following type of Downward Löwenheim-Skolem Theorems:

1'. For any structure A of countable signature, there is an elementary substructure B of A of cardinality \(<\aleph_2\) in terms of stationary logic.

2'. For any structure A of countable signature, there is an elementary substructure B of A of cardinality \(<2^{\aleph_0}\) in terms of stationary logic but only for formulas without free second order variables.

3'. For any structure A of countable signature, there is an elementary substructure B of A of cardinality \(<2^{\aleph_0}\) in terms of PKL logic (a variant of the stationary logic) in weak interpretation.

The reflection points \(<\aleph_2\) and \(<2^{\aleph_0}\) can be considered to be natural/necessary since the reflection down to \(<\aleph_2\) declares that \(\aleph_1\) strongly represents the situation of uncountability; the reflection down to \(<2^{\aleph_0}\) can be interpreted in the way that the reflection manifests that the continuum is very "rich".

The Downward Löwenheim-Skolem Theorems in terms of stationary logics can be also regarded as very natural principles: They can be characterized in terms of Diagonal Reflection Principles of Sean Cox.

Analyzing these three scenarios, we obtain the notion of Laver-generically large cardinals.

Existence of a Laver-generically supercompact cardinal

1''. for \(\sigma\)-closed pos implies 1'.;

2''. for proper pos implies 2'.; while the existence of a Laver-generically supercompact cardinal

3''. for ccc pos implies 3'.

The symmetry of the arguments involved suggests the possibility that the trichotomy might be a set-theoretic multiversal necessity.

If time allows, I shall also discuss about the reflection of non-metrizability of topological spaces, Rado's Conjecture and Galvin's Conjecture in connection with the reflection properties in 1., 2. and 3.

Most of the results to be presented here are obtained in a joint work with Hiroshi Sakai and André Ottenbreit Maschio Rodrigues.]]>

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This is a joint work with Tadek Dobrowolski and Mikołaj Krupski. The preprint containing these results can be found here: https://arxiv.org/abs/2002.07423

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More details can be found in the preprints (written jointly with Igor Protasov):

https://arxiv.org/abs/2004.01979

https://arxiv.org/abs/2002.08800]]>

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(i) the topology of \(X\) is generated by a metric d such that any two sets \(A, B\) of \(\mathcal{C}\) are parallel;

(ii) the cover \( \mathcal{C}\) is disjoint, lower semicontinuous and upper semicontinuous.

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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.]]>

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- Examples when usual destruction (that is, when \(\dot{X}\) required to be infinite only) implies \(+\)-destruction, and when it does not.
- Characterization of those Borel ideals which can be \(+\)-destroyed, in particular, we will see that if \(\mathcal{I}\) can be \(+\)-destroyed then the associated Mathias-Prikry forcing \(+\)-destroys it.
- Characterization of those analytic P-ideals which are \(+\)-destroyed by the associated Laver-Prikry forcing.

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For a continuum \(X\), let \(F_n(X)\) be the hyperspace of all nonempty subsets of \(X\)with at most \(n\)-points. The space \(F_n(X)\) is called the n'th-symmetric product.

In [1] it was proved that if \(X\)is a cone, then its hyperspace \(F_n(X)\) is also a cone.

During my talk I will discuss the converse problem. I will prove that if \(X\)is a locally connected curve, then the following conditions are equivalent:

- \(X\)is a cone,
- \(F_n(X)\) is a cone for some \(n\ge 2\),
- \(F_n(X)\) is a cone for each \(n\ge 2\).

[1] A. Illanes, V. Martinez-de-la-Vega, Symmetric products as cones, Topology Appl. 228 (2017), 36–46.

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In my talk the results cited above are proved and the mentioned question is answered under a (weaker) assumption \(\mathfrak b =\mathfrak c\).]]>

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The aim of the talk is to define basic notions from Fraisse theory, proof the main theorem and show some alternative way of looking at the construction of Fraisse limit.]]>

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It turns out that the positive answer follows from the existence of some large cardinals, while the counterexample can be found in the model of \(V=L\). ]]>

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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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