Set Theory

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.

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). 

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.

Abstract is attached

This is joint work with Miguel Moreno and Assaf Rinot.

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.

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.

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.

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

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.

 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.

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.

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.

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.

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 discuss some asymptotic variants of the club principle. Joint work with Shelah.

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.

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.

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.

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.

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.

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

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.

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.

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 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.

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 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 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.
 

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.

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

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.

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 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.

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 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

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.

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 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.

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.

Letcure Notes