# Set Theory

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Organizer(s)
Usual Time
Monday 13:00-15:00
Place
Area 502, Room 37

Upcoming Lectures
סמינרים | המחלקה למתמטיקה
Previous Lectures
- Omer Ben-Neria (HUJI)
Omer Ben-Neria (HUJI)
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In his PhD thesis, Luis Pereira has isolated two properties of sequences of regular cardinals (kappa_n)_n from Shelah's PCF theory, which are related to the possible consistency of  2^{aleph_omega} being greater or equal to aleph_{omega_1}, when aleph_omega is a strong limit cardinal.

The first property is the inexistence of a continuous tree-like scale on the product of regular cardinals kappa_n, n < omega. A scale <f_alpha : alpha < lambda> is said to be tree-like if for every alpha < beta and n < omega, if t_alpha(n) and t_\beta(n) are distinct, then so are t_alpha(m), t_beta(m) for all m >n.

The second and stronger assertion is the Approachable Free Subset Property (AFSP) which asserts that for almost every (i.e., modulo a club) internally approachable structure N of a sufficiently large H_\theta, of size |N| < kappa_m for some m, the set of supremums { delta^N_n = sup(N \cap kappa_n) : n < omega } has an infinite subset X which is free with respect to the functions in N. Namely, for every function g in N of some finite arity k, and every delta in X, delta does not belongs to the g-image of [ X - {delta} ]^k.

Gitik has shown that the existence of a sequence of regulars (kappa_n)_n for which there does not exist a continuous tree-like scale, is consistent relative to the existence of a measurable cardinal kappa of Mitchell order o(kappa) = kappa^{+2}.
In a recent join study with Dominik Adolf, we have shown that the existence of a sequence (kappa_n)_n for which the AFSP holds is consistent, and that both AFSP and the inexistence of continuous tree-like scales on some sequence (kappa_n)_n are equi-consistent with the existence of a singular cardinal kappa for which {o(mu) :  mu < kappa} is unbounded in kappa.

The goal of the talk is to present and discuss the two properties and their connection to PCF theory, and describe some of the central ideas in the recent results.

- Gabriel Fernandes (BIU)
Gabriel Fernandes (BIU)
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Jensen's covering theorem for Godel's constructible universe, L, says that if there is no non-trivial elementary embedding from L into L, then for every uncountable set of ordinals, X, there is a set, Y, such that Y is an element of L, |X| = |Y| and X is a subset of Y.

We will sketch the proof of the covering lemma for L.
- Menachem Magidor (HUJI)
Menachem Magidor (HUJI)
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I'll speak and I'll present some old results about weak diamond, uniformization and maybe some connections to Whitehead problem. In particular I'll present Woodin's elegant proof to the Devlin-Shelah equivalence of Weak diamond with 2^\aleph_0<2^\aleph_1.

- Jialiang He (BIU)
Jialiang He (BIU)
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This work is cooperated with Yinhe Peng and Liuzhen Wu.
I will present a few constructions for square of Menger group  problem in metrizable sense and generally sense in this talk.

Under cov(M)=c, for any n\geq 1, there is a subgroup G of Z^N such that G^n is Menger but G^{n+1} is not Menger.
Under cov(M)=d=cf(d)$, for any n\geq 1, there is a subgroup G of R such that G^n is Menger but G^{n+1} is not Menger. For any n\geq 1, There is a Menger subgroup G of R^{\omega_1} such that G^n is Menger butG^{n+1} is not Lindelof in ZFC. According to the known result, product of Menger subspace of Z^N is Menger in Miller model, this shows Menger group square problem is independent with ZFC in the metrizable sense. But for nonmetrizable topological group, the answer is no. - Uri Abraham (BGU) Uri Abraham (BGU) - we continue - Roy Shalev (BIU) Roy Shalev (BIU) - we continue - Uri Abraham (BGU) Uri Abraham (BGU) - we continue - Jing Zhang (BIU) Jing Zhang (BIU) - we continue - Uri Abraham (BGU) Uri Abraham (BGU) - we continue - Jing Zhang (BIU) Jing Zhang (BIU) - We'll talk about colorings of triples of w2 - Roy Shalev Roy Shalev - We shall present Fedorchuck's construction of a compact S-space of size 2^{w1}. If time permits, we shall show how it connects with the Moore-Mrowka problem. - Uri Abraham (BGU) Uri Abraham (BGU) - we continue - Tanmay Inamdar (BIU) Tanmay Inamdar (BIU) - we continue - Uri Abraham (BGU) Uri Abraham (BGU) - Results from the 2002 paper "Coding with Ladders a Well Ordering of the Reals" by Abraham and Shelah. Recoding is now available. - Tanmay Inamdar (BIU) Tanmay Inamdar (BIU) - I will talk about the titular paper by Komjath and Shelah. It is a short paper. - Tzoor Plotnikov (HUJI) Tzoor Plotnikov (HUJI) - Covering Neeman's proof - Istvan Juhasz (Renyi Institute) Istvan Juhasz (Renyi Institute) - - Mirna Dzamonja (UEA) Mirna Dzamonja (UEA) - A wide Aronszajn tree is a tree is size and height omega_1 but with no uncountable branch. Such trees arise naturally in the study of model-theoretic notions on models of size aleph_1 as well as in generalised descriptive set theory. In their 1994 paper devoted to various aspects of such trees, Mekler and Väänänen studied the so called weak embeddings between such trees, which are simply defined as strict-order preserving functions. Their work raised the question if under MA there exists a universal wide Aronszajn tree under such embeddings. We present a negative solution to this question, obtained in a paper to appear, joint with Shelah. We also discuss various connected notions and the history of the problem. - Matteo Viale (Torino) Matteo Viale (Torino) - We show that (assuming large cardinals) set theory is a tractable (and we dare to say tame) first order theory when formalized in a first order signature with natural predicate symbols for the basic definable concepts of second and third order arithmetic, and appealing to the model-theoretic notions of model completeness and model companionship. Specifically we develop a general framework linking generic absoluteness results to model companionship and show that (with the required care in details) a \Pi_2-property formalized in an appropriate language for second or third order number theory is forcible from some T extending ZFC + large cardinals if and only if it is consistent with the universal fragment of T if and only if it is realized in the model companion of T. Part (but not all) of our results are conditional to the proof of Schindler and Asperò that Woodin’s axiom (*) can be forced by a stationary set preserving forcing - Liuzhen Wu (Chinese Acad. Sciences, Beijing) Liuzhen Wu (Chinese Acad. Sciences, Beijing) - I will discuss a proof of the joint consistency of​ BPFA and \Delta_1-definablity of NS_{\omega_1}. Joint work with Stefan Hoffelner and Ralf Schindler. - Miguel Moreno (KGRC) Miguel Moreno (KGRC) - Filter reflection is an abstract version of stationary reflection. In this talk we will define filter reflection and different avatars of it. We will show the compatibility with large cardinals, forcing axioms, and V＝L. We will focus on the case when filter reflection holds and stationary reflaction fails, we will discuss how to force this case. We will also discuss the failure of filter reflection, how to force the failure and the requierements for it. If the time allows, some applications can be discussed. This is joint work with Gabriel Fernandes and Assaf Rinot. - Menachem Magidor (HUJI) Menachem Magidor (HUJI) - I'll speak about compactness problems for chromatic numbers of graphs. The main result will be a some what simplified proof of a theorem by Shelah, that a non reflecting stationary subset of a regular cardinal \lambda S\subseteq S^\lambda_kappa implies (under mild cardinal arithmetic assumptions) that there is a graph of size \lambda with chromatic number lambda , but every smaller cardinality subgraph has chromatic number <=kappa. - Alejandro Poveda (Universitat de Barcelona) Alejandro Poveda (Universitat de Barcelona) - (joint work with A. Rinot and D. Sinapova) In the previous talk, we introduced the notion of \Sigma-Prikry forcing and showed that many classical Prikry-type forcing which center on countable cofinalities fall into this framework. The aim of this talk is to present our iteration scheme for \Sigma-Prikry forcings. In case time permits, we will also show how to use this general iteration theorem to derive as a corollary the following strengthening of Sharon’s theorem: starting with \omega-many supercompact cardinals one can force a generic extension where Refl(<\omega,\kappa^+) holds and SCH_\kappa fails, for \kappa being a strong limit cardinal with cofinality \omega. The slides are now available. - Alejandro Poveda (Universitat de Barcelona) Alejandro Poveda (Universitat de Barcelona) - In a joint project with A. Rinot and D. Sinapova we introduce a class of notions of forcing which we call$\Sigma$-Prikry, and show that many of the known Prikry-type notions of forcing that centers around singular cardinals of countable cofinality are$\Sigma$-Prikry. Among these examples one may find Prikry forcing and its supercompact version, Gitik-Sharon forcing or the Extender Based Prikry forcing due to Gitik and Magidor. Our first result shows that there is a functor$\mathbb{A}(\cdot,\cdot)$which, given a$\Sigma$-Prikry poset$\mathbb P$and a name for a non-reflecting stationary set$\dot{T}$, yields a$\Sigma$-Prikry poset$\mathbb{A}(\mathbb{P},\dot{T})$that projects onto$\mathbb P$and kills the stationarity of$T$. Afterwards, we develop a viable iteration scheme for$\Sigma$-Prikry posets. In this talk I intend to give an overview of this theory and, if time permits, present the very first application of the method: namely, the consistency of a failure of the SCH_\kappa with$Refl(<\omega,\kappa^+)$, where$\kappa$is a strong limit singular cardinal of countable cofinality. The slides are now available. - Jing Zhang (BIU) Jing Zhang (BIU) - We discuss the existence of certain transformation functions turning pairs of ordinals into triples (or pairs) of ordinals, that allows reductions of complicated Ramsey theoretic problems into simpler ones. We will focus on the existence of various kinds of strong colorings. The basic technique is Todorcevic's walks on ordinals. Joint work with Assaf Rinot. - Menachem Magidor (HUJI) Menachem Magidor (HUJI) - I'll speak about the Friedman-Martin theorem that Borel determinacy can not be proved in Zermelo Set Theoy. (Namely, one needs reflection for getting it). Recoding is now available. - Menachem Kojman (BGU) Menachem Kojman (BGU) - Let p:[\kappa]^2\to \theta be a partition of all unordered pairs from a cardinal \kappa to \theta pieces. A coloring f:[\kappa]^2\to \lambda is *strong over *p if for every A\subseteq \kappa with |A|=\kappa there is some i=i(A)<\theta such that ran (f\restriction ([A]^2\cap p^{-1}(\{i\}))=\lambda. The partition symbol for asserting the existence of a strong \lambda-coloring on \kappa over a partition p is \kappa\not\longrightarrow_p[\kappa]^2_\lambda. In the talk we shall define more strong coloring symbols, like Pr_1 and Pr_0 and sketch the proofs of the following results: 1. Strong colorings over finite partitions exist in ZFC whenever they exist without partitions. 2. Instances of the GCH and of the SCH imply the existence of strong colorings over infinite partitions. 3. Whether for every countable partition of p:[\omega_1]^2\to \omega there is a strong \aleph_1-coloring over it, is independent over ZFC + \neg CH. These are joint results with Bill Chen and Juris Steprans. Recoding is now available. - Tanmay Inamdar (BIU) Tanmay Inamdar (BIU) - I will discuss the paper of Peng and Wu A Lindelof group with non-Lindelof Square'. I will focus on the oscillation results and the projection function. If time and patience permits, I will discuss some (unmotivated) slight strengthenings of their results in the partition calculus. - Tanmay Inamdar (BIU) Tanmay Inamdar (BIU) - I will discuss the paper of Peng and Wu A Lindelof group with non-Lindelof Square'. I will focus on the oscillation results and the projection function. If time and patience permits, I will discuss some (unmotivated) slight strengthenings of their results in the partition calculus. - Tanmay Inamdar (BIU) Tanmay Inamdar (BIU) - I will discuss the paper of Peng and Wu A Lindelof group with non-Lindelof Square'. I will focus on the oscillation results and the projection function. If time and patience permits, I will discuss some (unmotivated) slight strengthenings of their results in the partition calculus. - Tanmay Inamdar (BIU) Tanmay Inamdar (BIU) - I will discuss the paper of Shelah from the title and results which can be proved using similar techniques. A sample result is that one cannot have a chain of subsets of Omega2 strongly increasing modulo finite. The negative results we discuss can be contrasted with positive results of Koszmider on strong chains and results of Baumgartner on strongly almost disjoint families. The paper of Shelah is very short and relies only on a basic understanding of set theory (regular cardinals, closed unbounded sets). - Šárka Stejskalová (Institute of Mathematics, Prague) Šárka Stejskalová (Institute of Mathematics, Prague) - In the talk we will focus on compactness principles at the double successor of a regular cardinal kappa. We start by showing that if kappa^{<kappa} =kappa and lambda>kappa is a weakly compact cardinal, then in the Mitchell model V[M(kappa,lambda)] the tree property at kappa^{++} is indestructible under all kappa^+-cc forcing notions which live in the intermediate submodel V[Add(kappa,lambda)]. This result has direct applications for Prikry-style forcing notions and hence for the tree property at the double successor of a singular strong limit cardinal (it simplifies existing results and can be used to prove new results). Then we will discuss stationary reflection and its variants and the indestructibility under kappa^+-cc forcing notions. - Šárka Stejskalová (Institute of Mathematics, Prague) Šárka Stejskalová (Institute of Mathematics, Prague) - In the talk we will focus on compactness principles at the double successor of a regular cardinal kappa. We start by showing that if kappa^{<kappa} =kappa and lambda>kappa is a weakly compact cardinal, then in the Mitchell model V[M(kappa,lambda)] the tree property at kappa^{++} is indestructible under all kappa^+-cc forcing notions which live in the intermediate submodel V[Add(kappa,lambda)]. This result has direct applications for Prikry-style forcing notions and hence for the tree property at the double successor of a singular strong limit cardinal (it simplifies existing results and can be used to prove new results). Then we will discuss stationary reflection and its variants and the indestructibility under kappa^+-cc forcing notions. - Ari Meir Brodsky (BIU) Ari Meir Brodsky (BIU) - We present an introduction to the subject of constructing Souslin trees and the challenges involved, focusing on our uniform construction of Souslin trees (at arbitrary uncountable regular cardinals) using the "microscopic approach". We will show that the constructions following this approach can be carried out even in the absence of diamond, providing a significant improvement at the level of a strongly inaccessible cardinal. This is joint work with Assaf Rinot. - Jing Zhang (BIU) Jing Zhang (BIU) - Compactness principles often times imply guessing principles, for example, if kappa is measurable, then the diamond principle holds at kappa. Though small uncountable cardinals (like omega_2) is not a large cardinal, there are many compactness principles that can consistently hold at omega_2. A theorem of Shelah states in the model where GCH holds and omega_2 is sufficiently’’ compact, then the diamond principle holds at the points in omega_2 having cofinality omega_1. We will demonstrate a scenario where omega_2 is still pretty compact’’, but the diamond principle as above fails in a rather severe way. This shows Shelah’s theorem is optimal in some sense. Some relationships between our model and the model of GCH + aleph_2-Souslin hypothesis (if it exists) will also be discussed. - Roy Shalev (BIU) Roy Shalev (BIU) - Assuming the existence of a Souslin tree one can define a topology over the tree to get an Ostaszewski space. In this talk, we shall show how the existence of a Souslin tree gives rise to a topology on w1 which will form a "linear" Ostaszewski space. - Peter Komjath (ELTE) Peter Komjath (ELTE) - After an introduction to the basic facts on the coloring number, we give some applications to problems that do not seem to be connected to this notion. - Andrea Vaccaro (BGU) Andrea Vaccaro (BGU) - In this talk I will present an application of forcing techniques to C*-algebras, objects coming from functional analysis. A C*-algebra is a norm-closed, self-adjoint subalgebra of B(H), the algebra of all linear bounded operators on a complex Hilbert space H. The Calkin algebra Q(H), defined as the quotient of B(H) modulo the ideal of compact operators, is a C*-algebra which, for many good reasons, is considered the noncommutative analogue of the boolean algebra P(omega)/Fin. In this talk I will prove that, given any C*-algebra A, there is a ccc forcing E_A which forces the existence of an embedding of A inside Q(H). The benchmark for our construction is the analogous fact holding for boolean algebras: given any boolean algebra B, there is a ccc forcing E_B which forces the existence of an embedding of B inside P(omega)/Fin. While the definition of E_B is elementary, its adaptation to C*-algebras is fairly involved and it requires deep results from operator algebras. - Natasha Dobrinen (Denver) Natasha Dobrinen (Denver) - Abstract is attached - Shehzad Ahmed (BIU) Shehzad Ahmed (BIU) - Our goal is to finish the proof that an inaccessible Jonson cardinal lambda must be lambda x omega Jonsson. We begin by reviewing some properties of the iterated trace operator, and discussing how it interacts with certain sets coming from Jonsson models. From there, we will recall some important results from the previous talks and prove the desired results. - Shehzad Ahmed (BIU) Shehzad Ahmed (BIU) - In the previous talk, we showed that if lambda is an inaccessible Jonsson cardinal, then lambda must be Mahlo. In this talk, we will start by reviewing some basic ideas from the previous talk, and start developing the machinery required to prove Shelah's full result: If lambda is an inaccessible Jonsson cardinal, then lambda must be lambda x omega Mahlo. Along the way, we will discuss Shelah's rank ideals as well as related club guessing results. - Shehzad Ahmed (BIU) Shehzad Ahmed (BIU) - In Sh413, Shelah showed that an inaccessible Jonsson cardinal \kappa must be \kappa x \omega Mahlo. In this series of talks, we will provide a new proof of this result. The first part will focus on showing that an inaccessible Jonsson cardinal must be Mahlo. - Gabriel Fernandes (BIU) Gabriel Fernandes (BIU) - This is joint work with Miguel Moreno and Assaf Rinot. - Dominik Adolf (BIU) Dominik Adolf (BIU) - This is a continuation of the lecture from last week. - Dominik Adolf (BIU) Dominik Adolf (BIU) - We will show how the argument from the proof of the covering lemma for the core model can be utilized to analyze the PCF structure of certain products. Which products can be analyzed depends on the severity of anti-large cardinal assumptions. The greatest generality seems to be achieved if the Mitchell order of cardinals in the core model is bounded below the singular cardinal in question. This is joint work with Omer Ben-Neria. - Jing Zhang (BIU) Jing Zhang (BIU) - this is a continuation of our previous talk - Jing Zhang (BIU) Jing Zhang (BIU) - A typical problem studied in Rainbow Ramsey theory is: given a coloring of pairs of ordinals such that each color is used “not too many times”, is it possible to find a subset that is “rainbow”, namely the coloring restricted to the pairs of this subset is injective. Different problems arise by varying the parameters, for example, the “type” of the source cardinal (successor of regular/regular, limit singular/regular) or the exact meaning of “not too many times”. In this talk, we will survey the known results on different variations, with some attention paid to the connection with other combinatorial principles in set theory (eg. saturation of ideals, stationary reflection, various square principles). The ultimate goal is to motivate unsolved questions in this area. The talk will be self-contained. - Roy Shalev (BIU) Roy Shalev (BIU) - Last week, we formulated a weakening of Ostaszewski's club principle, and showed that it holds assuming the existence of a Souslin tree. In this talk, we shall show that our principle suffices for the construction of an Ostaszewski space. - Roy Shalev (BIU) Roy Shalev (BIU) - We formulate a weakening of Ostaszewski's club principle, and show that it holds assuming the existence of a Souslin tree. In the next talk, we shall show that our principle suffices for the construction of an Ostaszewski space. - Alejandro Poveda (Universitat de Barcelona) Alejandro Poveda (Universitat de Barcelona) - In this talk we shall present a proof of the consistency, modulo suitable large cardinals assumptions, of the following theory: "There is a strong limit cardinal \kappa with cof(\kappa)>\aleph_0 such that TP(\kappa^+) and TP(\kappa^{++}) hold and 2^\kappa is arbitrarily large" Here by arbitrarily large we mean that 2^\kappa can be any cardinal \gamma\geq \kappa^{++} with cof(\gamma)>\kappa.The proof relies on ideas of Sinapova, Unger and Friedman-Honzik-Stejskalová and provides a generalization to two results of Sinapova and Friedman-Honzik-Stejskalová, respectively. - Alejandro Poveda (Universitat de Barcelona) Alejandro Poveda (Universitat de Barcelona) - In this talk we shall present a proof of the consistency, modulo suitable large cardinals assumptions, of the following theory: "There is a strong limit cardinal \kappa with cof(\kappa)>\aleph_0 such that TP(\kappa^+) and TP(\kappa^{++}) hold and 2^\kappa is arbitrarily large" Here by arbitrarily large we mean that 2^\kappa can be any cardinal \gamma\geq \kappa^{++} with cof(\gamma)>\kappa.The proof relies on ideas of Sinapova, Unger and Friedman-Honzik-Stejskalová and provides a generalization to two results of Sinapova and Friedman-Honzik-Stejskalová, respectively. - Alejandro Poveda (Universitat de Barcelona) Alejandro Poveda (Universitat de Barcelona) - In this talk we shall present a proof of the consistency, modulo suitable large cardinals assumptions, of the following theory: "There is a strong limit cardinal \kappa with cof(\kappa)>\aleph_0 such that TP(\kappa^+) and TP(\kappa^{++}) hold and 2^\kappa is arbitrarily large" Here by arbitrarily large we mean that 2^\kappa can be any cardinal \gamma\geq \kappa^{++} with cof(\gamma)>\kappa.The proof relies on ideas of Sinapova, Unger and Friedman-Honzik-Stejskalová and provides a generalization to two results of Sinapova and Friedman-Honzik-Stejskalová, respectively. - Alejandro Poveda (Universitat de Barcelona) Alejandro Poveda (Universitat de Barcelona) - In this talk we shall present a proof of the consistency, modulo suitable large cardinals assumptions, of the following theory: "There is a strong limit cardinal \kappa with cof(\kappa)>\aleph_0 such that TP(\kappa^+) and TP(\kappa^{++}) hold and 2^\kappa is arbitrarily large" Here by arbitrarily large we mean that 2^\kappa can be any cardinal \gamma\geq \kappa^{++} with cof(\gamma)>\kappa.The proof relies on ideas of Sinapova, Unger and Friedman-Honzik-Stejskalová and provides a generalization to two results of Sinapova and Friedman-Honzik-Stejskalová, respectively. - Jialiang He (BIU) Jialiang He (BIU) - We will finish the proof from the last meeting. - Gabriel Fernandes (BIU) Gabriel Fernandes (BIU) - Local club condensation (LCC) is an abstract formulation of a condensation property that canonical L-like models have. It was introduced by Friedman and Holy. Diamond^{sharp} is a strengthening of Jensen's diamond principle which was introduced by Devlin. We shall show that diamond^{sharp} follows from LCC and the existence of a suitable Delta_1 definable well order, and then present an application of diamond^{sharp} to Generalized Descriptive Set Theory (GDST). This is joint work with Miguel Moreno and Assaf Rinot - Jialiang He (BIU) Jialiang He (BIU) - The Minimal Tower Problem was one of most famous question in Cardinal Invariants. We will present a combinatorial argument of this proof, which without using model theory and forcing, motivated by Malliaris and Shelah's proof. - Asger Törnquist (Københavns Universitet) Asger Törnquist (Københavns Universitet) - I will give an overview of the developments in the past 5 years regarding mad families. We'll study families of subsets of the natural numbers, and say that such a family is almost disjoint if any two distinct elements intersect finitely. The Axiom of Choice implies the existence of infinite almost disjoint family which is maximal under inclusion. Mathias proved in the late 1960s that it is consistent with ZF+DC that there are no mad families. He needed a Mahlo cardinal to do this. In 2014 I showed that the classical Solovay-Lévy model has no infinite mad families, and shortly thereafter, in 2016, Horowitz and Shelah showed that you don't even need an inaccessible to get a model of ZF+DC+no infinite mad families. A wealth of related questions have also been settled, most recently, I have shown with David Schrittesser that "All sets are Ramsey"+"Ramsey uniformization" implies "no infinite mad families ". I'll also discuss open problems. The talk will not assume any prior knowledge of mad families. - Ralf Schindler (Universität Münster) Ralf Schindler (Universität Münster) - In the 1970'ies, Bukowský identified a beautiful and handy criterion for when V is a forcing extension of a given inner model, which proved very useful recently in set theoretical geology. In the 1990'ies, Woodin isolated his extender algebra which makes use of a large cardinal, a Woodin cardinal. It turns out that Bukowský's theorem and Woodin's extender algebra may be presented in a uniform fashion - one proof and one forcing gives both results. We will present the proof and then discuss its application in inner model theoretic geology. This is joint work with Grigor Sargsyan and Farmer Schlutzenberg. - Miguel Moreno (BIU) Miguel Moreno (BIU) - During this talk we will discuss where in the generalized Borel-reducibility hierarchy are the isomorphism relation of first order complete theories. These theories are divided in two kind:classifiable and non-classifiable. To study the classifiable theories case is needed the use of Ehrenfeucht-Fraïssé games. On the other hand the study of the non-classifiable theories is done by using colored trees. The goal of the talk is to see the classifiable theories case and start the non-classifiable theories case by proving that it is possible to map every element of the generalized Baire, f, into a colored tree, J(f), such that; for every f and g elements of the generalized Baire space, J(f) and J(g) are isomorphic as colored trees if and only if f and g coincide on a club. - Jialiang He (BIU) Jialiang He (BIU) - An infinite cover of a topological space is an w-cover if every finite subset of this space is contained in some member of the cover, and the whole space is not a member the cover. A cover of a topological space is a gamma-cover if every point of this space belongs to all but finitely many members of this cover. A gamma-space is a space in which every open w-cover contains a gamma-cover. In this talk, we will present the details how to construct it. - Sandra Müller (KGRC) Sandra Müller (KGRC) - We will study infinite two player games and the large cardinal strength corresponding to their determinacy. For games of length$\omega$this is well understood and there is a tight connection between the determinacy of projective games and the existence of canonical inner models with Woodin cardinals. For games of arbitrary countable length, Itay Neeman proved the determinacy of analytic games of length$\omega \cdot \theta$for countable$\theta
\> \omega$from a sharp for$\theta$Woodin cardinals. We aim for a converse at successor ordinals. In joint work with Juan P. Aguilera we showed that determinacy of$\boldsymbol\Pi^1\_{n+1}$games of length$\omega^2$implies the existence of a premouse with$\omega+n$Woodin cardinals. This generalizes to a premouse with$\omega+\omega$Woodin cardinals from the determinacy of games of length$\omega^2$with$\Game^{\mathbb{R}}\boldsymbol\Pi^1\_1$payoff. If time allows, we will also sketch how these methods can be adapted to, in combination with results of Nam Trang, obtain$\omega^\alpha+n$Woodin cardinals for countable ordinals$\alpha$and natural numbers$n$from the determinacy of sufficiently long projective games. - Dominik Adolf (BIU) Dominik Adolf (BIU) - An uncountable cardinal κ is Jonnson if only if the set of proper subsets of κ that are of cardinality κ is stationary. Though this property has large cardinal strength it is not at all clear that Jonnson cardinals do in fact need to be large in the obvious sense. For example, it is known that Jonsson cardinals can be singular. In this talk we will use the methods of Inner Model Theory to show that, given the assumption that the least singular cardinal is Jonsson, there is a canonical model with a strong cardinal together with a class of Silver indiscernibles for this model. (The proof presented will make some simplifying assumptions.) Time permitting, we may discuss approaches to extend this result to show the existence of inner models with Woodin cardinals and more. - Daniel T. Soukup (KGRC) Daniel T. Soukup (KGRC) - Sierpinski's now classical result states that there is an edge 2-colouring of the complete graph on aleph1 vertices so that there are no uncountable monochromatic subgraphs. In the 1970s, Erdos, Galvin and Hajnal asked what other graphs with large chromatic number admit similar edge colourings i.e., with no 'large' monochromatic subgraphs. We plan to review some recent advances on this problem and in particular, connect the question to Shelah's ladder system uniformization theory. - Piotr Szewczak, Cardinal Stefan Wyszyński University in Warsaw, Poland Piotr Szewczak, Cardinal Stefan Wyszyński University in Warsaw, Poland - The theory of selection principles deals with the possibility of obtaining mathematically significant objects by selecting elements from sequences of sets. The studied properties mainly include covering properties, measure- and category-theoretic properties, and local properties in topological spaces, especially functions spaces. Often, the characterization of a mathematical property using a selection principle is a nontrivial task leading to new insights on the characterized property. I will give an overview of this theory and, if time permits, I will present some results obtained jointly with Boaz Tsaban and Lyubomyr Zdomskyy. - Miguel Moreno (BIU) Miguel Moreno (BIU) - Last week, we gave a detailed proof of Lemma 1.13 from the notes: http://u.math.biu.ac.il/~morenom3/GDST-2018.pdf This week, we shall continue, proving that, if V=L, then$\kappa$-Borel* class is equal to the$\Sigma1^ 1(\kappa)$class. - Miguel Moreno (BIU) Miguel Moreno (BIU) - After introducing the notions of$\kappa$-Borel class,$\kappa$-$\Delta_1^1$class,$\kappa$-Borel^* class we saw some subset relations between them in the previous talk ( http://u.math.biu.ac.il/~morenom3/GDST-2018.pdf ). We finished the previous talk with a sketch of the proof of: if V=L, then$\kappa$-Borel* class is equal to the$\Sigma1^ 1(\kappa)$class. We will see this proof in complete detail, starting from the key lemma, Lemma 1.13 on the notes. - Miguel Moreno (BIU) Miguel Moreno (BIU) - After introducing the notions of$\kappa$-Borel class,$\kappa$-$\Delta_1^1$class,$\kappa$-Borel^* class in the previous talk ( http://u.math.biu.ac.il/~morenom3/GDST-2018.pdf ), in this talk, we will show the relation between this classes. In descriptive set theory the Borel class, the$\Delta_1^1$class, the Borel* class are the same class, this doesn't hold in the generalized descriptive set theory, in particular under the assumption V=L the Borel* class is equal to the$\Sigma1^ 1$class. - Asaf Karagila (UEA) Asaf Karagila (UEA) - How big can countable unions of countable sets be? Assuming the axiom of choice, countable. Not assuming the axiom of choice, it is not hard to arrange situation where there are many incomparable cardinals which are the countable union of countable sets. But none of them are "particularly large". While a countable union of countable sets can at most be mapped onto \omega_1, its power set can be made much larger. We prove an old (and nearly forgotten) theorem of Douglass Morris, that it is consistent that for every \alpha there is a set which is a countable union of countable sets, but its power set can be mapped onto \alpha. - Miguel Moreno (BIU) Miguel Moreno (BIU) - This is the first of many of talks in which an overview of the Borel-reducibility hierarchy in the generalized Baire space will be given. The aim of this talk is to introduce the notions of$\kappa$-Borel class,$\kappa$-$\Delta_1^1$class,$\kappa$-Borel^* class, and show the relation between these classes. - Yair Hayut (HUJI) Yair Hayut (HUJI) - We discuss a joint work with Unger about stationary reflection at subcompact cardinals and the consistency of stationary reflection at successor of singular cardinal from a$Pi^1_1$-$kappa^+$-subcompact cardinal. - JiaLiang He (BIU) JiaLiang He (BIU) - We will address preservation theorems under Cohen forcing for various selection covering properties (Menger, Hurewicz, Rothberger and Gamma property). - JiaLiang He (BIU) JiaLiang He (BIU) - We will introduce four selection covering properties (Menger, Hurewicz, Rothberger and Gamma property), and consider preserving results on these properties under Cohen forcing. - David Fernández-Bretón (KGRC) David Fernández-Bretón (KGRC) - We will explore some (recent and not so recent; some positive, some negative) Ramsey-type results (each of which is due to some subset of the set {Komj\'ath, Hindman, Leader, H.S. Lee, P. Russell, Shelah, D. Soukup, Strauss, Rinot, Vidnyánszky, myself}) where abelian groups are coloured, and one attempts to obtain monochromatic sets defined in terms of the group structure. We will focus specifically on two families of very recent results: the first one concerns colouring groups with uncountably many colours, attempting to obtain finite monochromatic FS-sets; the second one concerns colouring groups (most of the time, our group of interest is the real line$\mathbb R$with its usual addition) with finitely many colours, attempting to obtain countably infinite monochromatic sumsets. - Chris Lambie-Hanson (BIU) Chris Lambie-Hanson (BIU) - In this talk, we will continue the investigation of questions involving Chang's Conjectures and the existence of certain club-increasing sequences. We begin by finishing our introduction of P_max forcing and sketching proofs of the facts that comprise the basic analysis thereof. We then present Larson's P_max variation that can be used to force the existence of a club-increasing sequence of length omega_2 in omega^omega over a model of AD^+ + V=L(A,R). - Chris Lambie-Hanson (BIU) Chris Lambie-Hanson (BIU) - The consistency of the Chang's Conjecture (CC) variant (aleph_{omega + 1}, aleph_omega) ->> (aleph_2, aleph_1) is a major open question. A combinatorial consequence of this instance of CC is the existence of a certain strongly increasing sequence, of length aleph_2, of functions from omega to some fixed ordinal below omega_2 (we are calling such a sequence a club-increasing sequence). In response to a question from the speaker, Paul Larson proved the consistency of the existence of a club-increasing sequence of length aleph_2 using a P_max forcing variation. In this talk, we will prove some basic results about club-increasing sequences, including the facts that their existence follows from the relevant instance of CC and that club-increasing sequences of length omega_n do not exist for n at least 4. We will then give an introduction to P_max forcing, in preparation for a future talk in which we will present a sketch of Larson's proof. - William Chen (BGU) William Chen (BGU) - We introduce a new cardinal invariant related to stick, prove some basic facts about it, and ask some open questions. Joint work with Geoff Galgon. - Ari Brodsky (Ariel University) Ari Brodsky (Ariel University) - We work with a$\lambda$-frame, which is an abstract elementary class endowed with a collection of basic types and a non-forking relation satisfying certain natural properties with respect to models of cardinality$\lambda$. We will show that assuming the diamond axiom, any basic type admits a non-forking extension that has a uniqueness triple. Prior results of Shelah in this direction required either some form of diamond at two consecutive cardinals, or a constraint on the number of models of size$\lambda$. This is joint work of with Adi Jarden. - Yair Hayut (TAU) Yair Hayut (TAU) - Strongly compact cardinals are characterized by the property that any$\kappa$-complete filter can be extended to a$\kappa$-complete ultrafilter. When restricting the cardinality of the underlying set, we obtain a nontrivial hierarchy. For example, when requiring the extension property to hold only for filters on$\kappa$, we obtain Gitik's$\kappa$-compact cardinals, which are known to be consistently weaker than$\kappa$being$\kappa^+$-strongly compact. In this talk I will focus on the level by level connection between the filter extension property and the compactness for$L_{\kappa,\kappa}$. Using the compactness, I will show that if$\kappa$is$\kappa$-compact then$\square(\kappa^{+})$-fails. - Assaf Rinot Assaf Rinot - In the 1970's, consistent examples of k-cc posets whose square is not k-cc were constructed by Laver, Fleissner, and Galvin. Later on, ZFC examples were constructed by Todorcevic, Shelah and others. The hardest case, being k=w2, was resolved by Shelah in 1997. In this work, we obtain analogous results for k-Knaster posets. Among others, for any successor cardinal k, we produce a ZFC example of a k-Knaster poset whose w-power is not k-cc. To do so, we introduce a new coloring principle, and establish the existence of various instances of it. We also introduce a new cardinal invariant for k, denoted chi(k), that, roughly speaking, measures how far k is from being weakly compact. It is proved that by forcing over a model with a weakly compact cardinal k, chi(k) could be made equal to any prescribed regular cardinal <= k. Further byproducts of this work show that the main results of [1] and [2] are sharp. This is joint work with Chris Lambie-Hanson. [1] A. Rinot, Transforming rectangles into squares, with applications to strong colorings, Adv. Math., 231(2): 1085-1099, 2012. [2] A. Rinot, Complicated colorings, Math. Res. Lett., 21(6): 1367–1388, 2014. - Ashutosh Kumar (HUJI) Ashutosh Kumar (HUJI) - We'll continue the discussion on some asymptotic variants of the club principle - the limsup versions. Joint work with Shelah. Here's the preperint: http://www.math.huji.ac.il/~akumar/svcp.pdf - Ashutosh Kumar (HUJI) Ashutosh Kumar (HUJI) - We'll discuss some asymptotic variants of the club principle. Joint work with Shelah. - Michael Megrelishvili (BIU) Michael Megrelishvili (BIU) - A circular order on a set is, intuitively speaking, a linear order which has been bent into a circle". In the first part we give necessary background, examples and motivation. In the second part we present some applications (joint results with E. Glasner) in symbolic dynamical systems and topological groups. We study ultra-homogeneous actions on circularly ordered sets and prove a "circular" analog of V. Pestov's well known result about ultra-homogeneous actions on linearly ordered sets. - Spencer Unger (TAU) Spencer Unger (TAU) - Continuing work from the previous talk, we sketch the proofs of some definable versions of Hall's matching theorem. Then we apply them to get various geometric paradoxes with definable pieces including our recent Borel circle squaring result with Andrew Marks. - Spencer Unger (TAU) Spencer Unger (TAU) - In recent years, there has been a resurgence in interest in the extent to which geometrical paradoxes can be done with definable pieces. A striking example of this is Dougherty and Foreman's solution to a problem of Marcewski: The Banach-Tarski paradox is possible using Baire measurable pieces. We survey some recent results in this area including joint work with Andrew Marks and Clinton Conley. - William Chen (BGU) William Chen (BGU) - Last week, we presented an approach for constructing a Dowker space of size contniuum, ending up with a statement of a lemma that would yield such a space. In this talk, we shall prove this lemma. Lecture notes may be found in here. - William Chen (BGU) William Chen (BGU) - A topological space is said to be Dowker if it is normal but its product with the unit interval is not normal. In this lecture, we shall present a construction, due to Balogh, of a Dowker space of size continuum. Lecture notes may be found in here. - Eilon Bilinsky (TAU) Eilon Bilinsky (TAU) - We construct a model of ZF with an uncountable set of reals having a unique condensation point. This answers a question of Sierpinski from 1918. - Chris Lambie-Hanson (BIU) Chris Lambie-Hanson (BIU) - We continue our pair of talks on connections between square principles, trees with ascent paths, and strong chain conditions. In the previous talk, we discussed trees with ascent paths; we turn our attention this week to chain conditions. In particular, we will prove that, if$\kappa > \aleph_1$is a regular cardinal and$\square(\kappa)$holds, then: 1) There is a$\kappa$-Knaster poset$\mathbb{P}$such that$\mathbb{P}^\omega$is not$\kappa$-c.c. 2) There is a$\kappa$-Knaster poset$\mathbb{P}$that is not$\kappa$-stationarily layered. This talk will rely only minimally on material from the previous talk, and the results are joint work with Philipp Lücke. - Chris Lambie-Hanson (BIU) Chris Lambie-Hanson (BIU) - Two topics of interest in modern set theory are the productivity of chain conditions and the existence of higher Aronszajn trees. In this talk, we discuss generalizations of both of these topics and their connections with various square principles. In particular, we will prove that, if$\kappa$is a regular uncountable cardinal and$\square(\kappa)$holds, then: 1) for all regular$\lambda < \kappa$, there is a$\kappa$-Aronszajn tree with a$\lambda$-ascent path; 2) there is a$\kappa$-Knaster poset$\mathbb{P}$such that$\mathbb{P}^{\aleph_0}$is not$\kappa$-c.c. Time permitting, we will also present a complete picture of the relationship between the existence of special trees and the existence of Aronszajn trees with ascent paths at the successor of a regular cardinal. This is joint work with Philipp Lücke. - Yair Hayut (TAU) Yair Hayut (TAU) - In this talk I will define the notion of Magidor Cardinal (\omega bounded Jonsson cardinal) which is a generalization of Jonsson cardinal. I will show that the analog of Jonsson filter for Magidor cardinals is inconsistent with ZFC. This lecture is based on a joint work with Shimon Garti and Saharon Shelah - Tom Benhamou (TAU) Tom Benhamou (TAU) - We shall present a new method for obtaining the tree property at w2 from the consistency of a weakly compact cardinal. The method is due to Stejskalova, and uses Grigorieff forcing. - Thilo Weinert (BGU) Thilo Weinert (BGU) - the abstract may be found in here. Ari Brodsky (BIU) Ari Brodsky (BIU) We address the question whether, assuming the generalized continuum hypothesis (GCH), the existence of a k-Aronszajn tree entails the existence of a distributive one. Biblography: see here. - Ashutosh Kumar (HUJI) Ashutosh Kumar (HUJI) - Fremlin asked if the null ideal restricted to a non null set of reals could be isomorphic to the non stationary ideal on omega_1. Eskew asked if the null and the meager ideal could both be somewhere countably saturated. We'll show that the answer to both questions is yes. Joint work with Shelah. - Ashutosh Kumar (HUJI) Ashutosh Kumar (HUJI) - Fremlin asked if the null ideal restricted to a non null set of reals could be isomorphic to the non stationary ideal on omega_1. Eskew asked if the null and the meager ideal could both be somewhere countably saturated. We'll show that the answer to both questions is yes. Joint work with Shelah. - Thilo Weinert (BGU) Thilo Weinert (BGU) - This is part 2 of last week's talk. - Thilo Weinert (BGU) Thilo Weinert (BGU) - We are going to continue the analysis of generalised scattered orders, proving the result described towards the end of Chris Lambie-Hanson’s talk. This states that consistently, for every sigma-scattered linear ordering there is a colouring of its pairs in black & white such that every triple contains a white pair and every copy of one of the following order-types contains a black pair: • omega_1^omega • (omega_1^omega)^* • omega_1 * (omega * omega^*)^omega • omega_1^* * (omega * omega^*)^omega • (omega * omega^*)^omega * omega_1 • (omega * omega^*)^omega * omega_1^* This generalises a 46-year-old Theorem of Erdős & Rado about ordinals. A sufficient hypothesis implying this theorem is the existence of a colouring of the pairs of omega_1 * omega in black & white such that every triple contains a black pair and every subset of full order-type contains a white one. Time permitting we may present a proof that stick = b = Aleph_1 implies the existence of such a colouring. Here b is the unbounding number and stick = Aleph_1 is a weakening of the club principle which was considered by Baumgartner 41 years ago, named by Broverman, Ginsburg, Kunen & Tall two years thereafter and twenty years ago reconsidered as a cardinal characteristic by Fuchino, Shelah & Soukup. - Chris Lambie-Hanson (BIU) Chris Lambie-Hanson (BIU) - The class of scattered linear orders, isolated by Hausdorff, plays a prominent role in the study of general linear orders. In 2006, Dzamonja and Thompson introduced classes of orders generalizing the class of scattered orders. For a regular cardinal kappa, they defined the classes of kappa-scattered and weakly kappa-scattered linear orders. For kappa = omega, these two classes coincide and are equal to the classical class of scattered orders. For larger values of kappa, though, the two classes are provably different. In this talk, we will investigate properties of these generalized scattered orders with respect to partition relations, in particular the extent to which the classes of kappa-scattered or weakly kappa-scattered linear orders of size kappa are closed under partition relations of the form tau -> (phi, n) for n < omega. We will show that, assuming kappa^{<kappa} = kappa, the class of weakly kappa-scattered orders is closed under all such partition relations while, for uncountable values of kappa, the class of kappa-scattered orders consistently fails to be closed. Along the way, we will prove a generalization of the Milner-Rado paradox and look at some results regarding ordinal partition relations. This is joint work with Thilo Weinert. - William Chen (BGU) William Chen (BGU) - This is an expository presentation following the paper "Minimality of non$\sigma$-scattered orders" by Ishiu and Moore. In the first part of the talk we will introduce the invariant$\Omega(L)$of a linear order$L$, and characterize$\sigma$-scattered linear orders in terms of this invariant. In the second part, we will prove under the forcing axiom$\mathsf{PFA}^+$that any linear order which is minimal with respect to embedding among the non$\sigma$-scattered orders must be either a real or Aronszajn type. - William Chen (BGU) William Chen (BGU) - This is an expository presentation following the paper "Minimality of non$\sigma$-scattered orders" by Ishiu and Moore. In the first part of the talk we will introduce the invariant$\Omega(L)$of a linear order$L$, and characterize$\sigma$-scattered linear orders in terms of this invariant. In the second part, we will prove under the forcing axiom$\mathsf{PFA}^+$that any linear order which is minimal with respect to embedding among the non$\sigma\$-scattered orders must be either a real or Aronszajn type.

- Assaf Rinot
Assaf Rinot
-

This is a continuation of last week's talk. This time, I shall prove that square(kappa) give rise to a partition of kappa into kappa many fat sets

- Assaf Rinot
Assaf Rinot
-

A subset F of a regular uncountable cardinal kappa is said to be fat iff for every club C in kappa, and every ordinal alpha<kappa, F\cap C contains a closed copy of alpha+1.
By a theorem of H. Friedman from 1974, every stationary subset of w1 is fat. In particular, w1 may be partitioned into w1 many pairwise disjoint fat sets.

In this talk, I shall prove that square(kappa) give rise to a partition of kappa into kappa many pairwise disjoint fat sets. In particular, the following are equiconsistent:

1. w2 cannot be partitioned into w2 many pairwise disjoint fat sets;
2. w2 cannot be partitioned into two disjoint fat sets;
3. there exists a weakly compact cardinal.

- Thilo Weinert (BGU)
Thilo Weinert (BGU)
-

We are going to prove a classical result of Baumgartner - the existence of a linear order type every uncountable subtype of which contains a copy of omega_1 yet fails to be the union of countably many well-ordered types.

The paper may be found in here.

- Thilo Weinert (BGU)
Thilo Weinert (BGU)
-

Abstract: We are going to prove a classical result of Baumgartner - the existence of a linear order type every uncountable subtype of which contains a copy of omega_1 yet fails to be the union of countably many well-ordered types.

The paper may be found in here.

- Thilo Weinert (BGU)
Thilo Weinert (BGU)
-

We are going to prove a classical result of Baumgartner - the existence of a linear order type every uncountable subtype of which contains a copy of omega_1 yet fails to be the union of countably many well-ordered types.

The paper may be found in here.

- Eran Stein
Eran Stein
-

Last week, we presented a construction scheme which is based on well-behaving delta-systems of finite subsets of w1. In this lecture, we shall present an application to the theory of uncountable trees.

The results are taken from the following paper.

- Eran Stein
Eran Stein
-

We present a construction scheme which is based on well-behaving delta-systems of finite subsets of w1, and use it to construct uncountable trees.

The results are taken from the following paper.

- Dani Livne
Dani Livne
-

We present Komjath's theorem that there exists a coloring of the real line in 2 colors such that for any uncountable subset A of reals, there exist 4 distinct elements a,b,c,d in A such that a+b gets color 0, and c+d gets color 1.

The results are taken from the following paper.

- Omri Marcus
Omri Marcus
-

We shall complete the verification of existence of obligatory graphs for infinite colouring numbers.
In our previous talk, we covered the case of graphs of regular cardinality. This time, we shall address graphs of singular cardinality.

- Omri Marcus
Omri Marcus
-

We show that every graph with inﬁnite colouring number has a well-ordering of its vertices that simultaneously witnesses its colouring number and its cardinality.

The lecture will be based on the following paper.

- Guy Kapon
Guy Kapon
-

A tree is said to be rigid if it has a trivial automorphism group. It is said to be homogeneous if any two nodes of the same level can be sent one to the other via an automorphism of the tree. In this talk, we shall present Larson's proof that the existence of a strongly homogeneous Souslin tree entails the existence of a strongly rigid Souslin tree.

- Eran Stein
Eran Stein
-

We shall resume the presention of Ostaszewski's construction of a perfectly normal, hereditarily separable, first countable, locally countable, locally compact, Hausdorff topological space in which every open set is either countable or co-countable.

- Eran Stein
Eran Stein
-

We shall present Ostaszewski's construction of a perfectly normal, hereditarily separable, first countable, locally countable, locally compact, Hausdorff topological space in which every open set is either countable or co-countable.

- Roy Shalev
Roy Shalev
-

This is part II of last week's talk.

- Roy Shalev
Roy Shalev
-

We shall present a recent theorem of Raghavan and Todorcevic that uses a Souslin tree to refute a particular generalization of the Erdos-Dushnik-Miller theorem.

- Ron Langberg
Ron Langberg
-

We shall present a construction (due to Milner and Shelah) of a very large graph which has no unfriendly 2-partition, and in which every vertex has infinite degree.

- Shahak Shama
Shahak Shama
-

We shall show that any ladder system on w1 induces a certain uncountable topological space, and then present sufficient conditions on the ladder system that makes the corresponding space into a Dowker space.

- Ari Brodsky (BIU)
Ari Brodsky (BIU)
-

We shall describe a construction of a Souslin tree, following our recent paper.

- Ari Brodsky (BIU)
Ari Brodsky (BIU)
-

We shall describe a construction of a Souslin tree, following our recent paper.

- Dani Livne
Dani Livne
-

Infinite trees and partition calculus (aka, Ramsey theory) are well-known to be intertwined. For instance, Ramsey theorem implies Konig's lemma that asserts that every infinite tree which is finitely branching has an infinite path.
In this talk, we shall deal with uncountable trees such as Souslin trees and Aronszajn trees, and show how to derive negative partition relations from them.

Lecture notes.

- Luis Pereira (Lisbon)
Luis Pereira (Lisbon)
-

The purpose of this talk is to present the main developments in Cardinal Arithmetic from 1960 to 1975. After a brief review of the basic independence results, we will review the basic definitions and results about ultrapowers and measurable cardinals and proceed to Scott's and Vopenka's results in Cardinal Arithmetic regarding measurable cardinals and singular cardinals of measurable cofinality. These results are generalizable to all singular cardinals of uncountable cofinality and this is what we will look at next. For that will start with the basic definitions and examples regarding the Galvin-Hajnal norm and finish with the application of the Galvin-Hajnal bound for families of almost disjoint functions to Cardinal Arithmetic.

- Piotr Szewczak (BIU and Cardinal Stefan Wyszyński University, Warsaw)
Piotr Szewczak (BIU and Cardinal Stefan Wyszyński University, Warsaw)
-

We introduce Menger and Hurewicz covering properties, which are generalizations of sigma-compactness. Menger and Hurewicz conjectured that, for subsets of the real line, the above properties were equivalent to sigma-compactness. Using topological and an elegant combinatorial method (coherent omission of intervals), we show (in ZFC) that they are false. We consider also stronger covering properties, relations between them and we give examples of such sets of reals. After that we obtain the solution to the Hurewicz problem: Is there in ZFC an example of set of reals which is Menger but not Hurewicz? Finally we show some results concerning behavior of Menger and Hurewicz properties in finite products.

The methods, proofs, and results, are mainly due to Tsaban and his collaborators. The last lecture will include new results, due to Tsaban and the speaker.
- Assaf Rinot
Assaf Rinot
-

Gray's combinatorial principle SD_k is a strong combination of Jensen's Square_k and Diamond(k^+) principles. This principle proved itself very useful in constructing uncountable graphs of counter-intuitive nature.

By a 35 year old theorem of Shelah, Square_k+Diamond(k^+) does not imply SD_k for regular uncountable cardinals k. In this talk, I will prove that they are equivalent whenever k is singular.

Bibliography

- Dani Livne
Dani Livne
-
- Shir Sivroni
Shir Sivroni
-

Which Isbell-Mrowka spaces spaces satisfy the star version of Menger’s covering property?

Following Bonanzinga and Matveev, this question is considered here from a combinatorical point of view.  We give an answer to a problem thay have stated,  and present some related open problems.

All is taken from this paper by Boaz Tsaban.
The slides are available here.

- Efrat Taub (BIU)
Efrat Taub (BIU)
-

Some combinatorial principles were invented by Jensen, in his analysis of Godel's constructible universe. One of them is Diamond^*.

We will introduce variants and generalizations of Diamond^* and discuss when these principles hold and when they do not hold.

- Michal Machura (BIU)
Michal Machura (BIU)
-

We shall present the P-hierarchy of ultrafilters, that was posed by Andrzej Starosolski.
The P-hierarchy of ultrafilters is one of many ways to classify ultrafilters on natural numbers and it is composed of ℵ1 disjoint classes P(α) where α is ordinal number <ω1. The class P(1) is just a class of principal ultrafilters. The class P(2) is composed of P-points,  which were isolated by Rudin in order to prove non-homogeneity of the remainder of Cech-Stone compactification of natural numbers. Next, in higher classes of P-hierarchy, one can find ultrafilters with more and more complicated structures.

In this talk, we will disscuss relations between classes P(α) of P-hierarchy and other special types of ultrafilters, including: Baumgartner’s I-ultrafilters, thin ultrafilters, summable ultrafilters, and van der Waerden ultrafilters.

- Nir Hakeyni (BIU)
Nir Hakeyni (BIU)
-

A topological space is called k-resolvable if it is the union of k many disjoint dense subsets. In this second lecture, we shall survey some of the results obtained throughout the years and record some open questions.

- Nir Hakeyni (BIU)
Nir Hakeyni (BIU)
-

A topological space is called resolvable if it is the union of two disjoint dense subsets. Since the concept was first defined and explored by Edwin Hewitt in 1943, much effort has been invested in obtaining general results concerning the resolvability or irresolvability of certain types of spaces, and in generating examples and counterexamples.
In the present lecture we will take a leisurely tour through the subject. We will discuss generalizations of the original concept, display some of the results obtained throughout the years and mention questions which are still open.

- Ari Brodsky (BIU)
Ari Brodsky (BIU)
-

We shall discuss generalizations of Ramsey's theorem to the context of trees of high chromatic number. A detailed abstract is available here.

Bibliograpy

Ashutosh Kumar (HUJI)
Ashutosh Kumar (HUJI)

Komjath has asked the following question: Let X be a subset of Euclidean space. Must there exist a subset Y of X such that X and Y have same outer measure and the distance between any two points in Y is irrational?

We'll show that this is true in dimension one. Our proof relies on some work of Gitik and Shelah on forcings with sigma ideals.

Chris Lambie-Hanson (HUJI)
Chris Lambie-Hanson (HUJI)

The amalgamation property is a topic of fundamental interest in model theory and is still imperfectly understood. In the 1980s, Grossberg asked a question, which remains open to this day, about the existence of a Hanf number for amalgamation in abstract elementary classes. We introduce a new class of structures, called well-colorings, and use them to give a partial answer to Grossberg’s question, significantly improving upon previous work of Baldwin, Kolesnikov, and Shelah. We shall start the talk by briefly discussing the relevant model-theoretic definitions (no prior model-theoretic knowledge will be assumed) and will then give proofs of the main results, which are entirely set-theoretic and combinatorial in nature and of interest in their own right. This is joint work with Alexei Kolesnikov.

Bibliography

Assaf Rinot
Assaf Rinot

We shall survey the history of the study of the productivity of the k-chain-condition in partial orders, topological spaces, and Boolean algebras. We shall address a conjecture that tries to characterize such a productivity in Ramsey-type language. For this, a new oscillation function for successor cardinals, and a new characteristic function for walks on ordinals will be proposed and investigated.

Bibliography

Assaf Rinot
Assaf Rinot

We shall present the notion of a Luzin Set, various generalizations, as well as applications to strong and not-so-strong colorings.

Dani Livne
Dani Livne

Dani shall present costructions (due to Hajnal) of Anti-Ramsey colorings which are not universal. That is, these colorings fail to embed particular finite patterns. Unlike Shelah's construction (that Michal presented), these construction will be carried in ZFC.

Assaf Rinot
Assaf Rinot

We shall present various concepts of being a "large" subset of w1.

Tomer Bauer
Tomer Bauer

Can you tell the present by knowing the future? That is, can there be a function f:[X]w-->X so that given a sequence <x0,x1,x2,...>, we would have xn=f(xn+1,xn+2,xn+3,....) for (almost) all n?

This type of problems was considered by Galvin, Erdos-Hajnal, Prikry, and Solovay in the 1960's and 1970's, and regained interest more recently in the study of generalized hat problems.

Tomer's talk will present this line of research.

Michal Machura
Michal Machura

Michal shall prove Shelah's thereom that the Continuum Hypothesis entails a coloring c:[w1]2-->w such that c`[A]2=w for every uncountable subset A of w1, and yet c admits no 3-sized set X on which c|[X]2 is one-to-one.

Lecture notes

Dani Livne
Dani Livne

We shall prove that for every infinite cardinal k, there exists a coloring c:[k]wX satisfying the following:

• c is 2-to-1;
• c restricted to  [A]w is not injective for every infinite A.

Letcure Notes

Dani Livne (BGU)
Dani Livne (BGU)

We shall provide sufficient conditions for the existence of a function f:[w1]2→w1 satisfying the following:

• f is 2-to-1;
• f restricted to any uncountable square [X]2 is not injective.

Lecture Notes

Yuval Hachatrian
Yuval Hachatrian

An L-space is a regular topological space which is hereditary Lindelof, but not separable. Yuval will present a sufficient condition for the existence of an L-space: a combination of an uncountable b-universal sequence, and an L-syndetic coloring give rise to such a space.

Lecture notes

Yuval Hachatrian
Yuval Hachatrian

Yuval will show how to read a b-universal binary sequence of length continuum from Kronecker's theorem on simultaneous diophantine approximation.

Lecture notes

Lidor Eldabah
Lidor Eldabah

Lidor will present a proof of Todorcevic's theorem stating that there is a continuous coloring of all triples of rational numbers in coutnably many colors, in such a way that for any topological copy C of the rationals and any possible color k, there exists a triple in C that is colored with the desired color k.

Lecture notes

לידור אלדבח
לידור אלדבח

Lidor will present a proof of Baumgartner's theorem stating that there is a coloring of all pairs of rational numbers in coutnably many colors, in such a way that for any topological copy C of the rationals and any possible color k, there exists a pair in C that is colored with the desired color k.

Lecture notes

אסף רינות
אסף רינות

במפגש הראשון דנו במשפט רמזי הסופי והאינסופי, ובדוגמא של שרפינסקי המראה כי ההכללה המתבקשת למקרה שאיננו בן מניה - איננה נכונה. דיברנו על סוגי צביעות המעידות על כשלון תופעות מסוג רמזי, ועל הגרסא האולטימטיבית של "צביעה חזקה", כמו גם, גרסאות אסימטריות.

דיברנו על שמורות מונים של מרחבים טופולוגיים, והשוונו בין המושגים: "בן מניה שתיים", "ספרבילי" ו"לינדלוף". הזכרנו שהמושגים שקולים בהקשר של מרחבים מטריים, ובחרנו להתמקד במקרה של מרחבים רגולריים. רמזנו שצביעות חזקות מאפשרות להגדיר מרחבים רגולריים המקיימים תכונה אחת, ולא את השניה: למשל מרחב רגולרי ספרבילי תורשתית, שאיננו לינדלוף. מנגד, הזכרנו כי הטענה כי "כל מרחב רגולרי ספרבילי תורשתית הוא לינדלוף" מתיישבת עם האקסיומות הרגילות של תורת הקבוצות.