# Set Theory

### Previous Lectures

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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

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.

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.

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.

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.

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:

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

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.

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.

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.

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.

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.

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.

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

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:

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

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.

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.

The paper may be found in here.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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

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

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.

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.

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.

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.

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.

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.

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.

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.

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.

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

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?

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.

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.

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

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.

We shall present various concepts of being a "large" subset of w_{1}.

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 <x_{0},x_{1},x_{2},...>, we would have x_{n}=f(x_{n+1},x_{n+2},x_{n+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 shall prove Shelah's thereom that the Continuum Hypothesis entails a coloring c:[w_{1}]^{2}-->w such that c``[A]^{2}=w for every uncountable subset A of w_{1}, and yet c admits no 3-sized set X on which c|[X]^{2} is one-to-one.

We shall prove that for every infinite cardinal k, there exists a coloring c:[_{k}]^{w}→_{X} satisfying the following:

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

We shall provide sufficient conditions for the existence of a function f:[w_{1}]^{2}→w_{1} satisfying the following:

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

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.

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

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.

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.

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.

- תאריך עדכון אחרון: 20/01/2019