Set Theory

Usual Time
Monday, 13:00-15:00
Place
Building 216, room 201
Upcoming Lectures
סמינרים | המחלקה למתמטיקה
- , Intersection of a sequence of outer models, part 2, Zhixing You (BIU)
Zhixing You (BIU)
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Place: Seminar room

We continue

Previous Lectures
- , Intersection of a sequence of outer models, part 1, Zhixing You (BIU)
Zhixing You (BIU)
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Place: Seminar room
 

The technique of taking the intersection of a sequence of outer models is understudied in the field of Mathematical Logic. With Assaf Rinot and Jiachen Yuan, we find it to be quite effective for constructing some interesting sum ultrafilters. In this talk, I will first illustrate how we use it to answer questions about indecomposable ultrafilters and $\delta$-complete uniform ultrafilters, and then extend some results in infinitary combinatorics. A sample result is that we construct a model in which there exists a weakly compact cardinal that carries an indecomposable ultrafilter but not measurable, thus answering an old question of Ketonen in the negative.

- , Splitting families of sets, part 2, Tal Kagalovsky (BGU)
Tal Kagalovsky (BGU)
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Place: Seminar room

We continue

- , Splitting families of sets, part 1, Tal Kagalovsky (BGU)
Tal Kagalovsky (BGU)
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Place: Seminar room

I will present several known results in the topic of splitting families of sets. Miller’s splitting theorem from 1937 was proven for finite n > 0 and ρ-uniform families where ρ is infinite. Under the assumptions of GCH and without the use of more modern tools, this approach is generalized here in order to prove several stronger theorems by Erdős and Hajnal.
Then a more modern approach by Kojman will be presented, where he used Shelah’s RGCH in order to eliminate the GCH from some of the theorems when ρ ≥ בω (ν). This was done using filtrations with respect to anti-monotone functions.
 

- , A perfect space whose square is not cmc, Shira Yadai (BIU)
Shira Yadai (BIU)
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Place: Seminar room

We present a consistent construction of a topological space that is perfect (all closed sets are G_delta) whose square is not countably metacompact.

This is joint work with Assaf Rinot

- , A special T(rho_1), Tanmay Inamdar (BGU)
Tanmay Inamdar (BGU)
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Place: Seaminar room

We give a criterion on a C-sequence which implies that the associated tree T(rho_1) is special. In particular such a tree exists in ZFC. This answers Question 2.2.18 from Todorcevic' walks book. We prove some related results.

This is joint work with Assaf Rinot

- , A topological Ramsey theorem, Tanmay Inamdar (BGU)
Tanmay Inamdar (BGU)
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Place: Building 203, room 221

We prove Galvin's conjecture: for every colouring of pairs of reals into finitely-many colours, there is a set homeomorphic to the rationals on which at most two colours are taken

- , On indecomposable ultrafilters at every successor of singular, part 6, Inbar Oren (HUJI)
Inbar Oren (HUJI)
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Place: Seminar room

We continue

- , On indecomposable ultrafilters at every successor of singular, part 5, Inbar Oren (HUJI)
Inbar Oren (HUJI)
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Place: Seminar room

We continue

- , On indecomposable ultrafilters at every successor of singular, part 4, Inbar Oren (HUJI)
Inbar Oren (HUJI)
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Place: Seminar room

We continue

- , On indecomposable ultrafilters at every successor of singular, part 3, Inbar Oren (HUJI)
Inbar Oren (HUJI)
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Place: Seminar room

We continue

- , On indecomposable ultrafilters at every successor of singular, part 2, Inbar Oren (HUJI)
Inbar Oren (HUJI)
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Place: Seminar room

We continue

- , On indecomposable ultrafilters at every successor of singular, part 1, Inbar Oren (HUJI)
Inbar Oren (HUJI)
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Place: Seminar room

Indecomposability of an ultrafilter is a weakening of a completeness of an ultrafilter. Ben-David and Magidor showed that it is possible to obtain an indecomposable ultrafilter at א_{ω+1}. In our work, we extend their result to obtain indecomposable ultrafilters at every successor of a singular cardinal. This result will be the focus of our talks.
We will also provide applications in two separate directions. Firstly, we will apply one of Shelah’s results to obtain a model where the compactness of chromatic numbers of graphs holds on every successor of a singular. We will also apply a proof from Raghavan and Shelah to show that for any fixed successor of a singular, it is consistent to have a small ultrafilter number at that cardinal. Secondly, we will give a few examples where Silver type theorems fail.

This is joint work with Sittinon Jirattikansakul and Assaf Rinot

- , Forcing with models as side conditions, part 7, Ido Feldman (BIU)
Ido Feldman (BIU)
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Place: Seminar room

We continue

- , Forcing with models as side conditions, part 6, Ido Feldman (BIU)
Ido Feldman (BIU)
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Place: Building 216, Room 132

We continue

- , Forcing with models as side conditions, part 5, Ido Feldman (BIU)
Ido Feldman (BIU)
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Place: Seminar room

We continue

- , Forcing with models as side conditions, part 4, Ido Feldman (BIU)
Ido Feldman (BIU)
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Place: Seminar room

We continue

- , Omega-bounded ladder systems, Assaf Rinot
Assaf Rinot
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Place: Seminar room

Leiderman and Szeptycki proved that a single Cohen real introduces a ladder system L over w1 for which the corresponding space X_L is not a Delta-space. They asked whether there is a ZFC example of a ladder system L over some cardinal kappa for which X_L is not countably metacompact, in particular, not a Delta-space. In a paper from August, we gave an affirmative answer with kappa=cf(beth_{w+1}), but we got the feedback that the biggest interest is ladder systems consisting of ladders of order-type omega. In this talk, we'll prove that if aleph_w is a strong limit, then this extra requirement can be satisfied as well.

This is joint work with Rodrigo Rey Carvalho and Tanmay Inamdar.

- , Forcing with models as side conditions, part 3, Ido Feldman (BIU)
Ido Feldman (BIU)
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Place: Seminar room

We continue

- , The vanishing levels of a tree, Zhixing You (BIU)
Zhixing You (BIU)
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Place: Seminar room

We prove the consistency of the existence of a kappa-Souslin tree whose set of vanishing levels is a club in a model where all nontrivial C-sequences over kappa are trivial. This is joint work with Assaf Rinot and Jiachen Yuan.

- , Diamond on Kurepa trees, part 2, Ziemek Kostana (BIU)
Ziemek Kostana (BIU)
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Place: Building 216, Room 132

We continue

- , Forcing with models as side conditions, part 2, Ido Feldman (BIU)
Ido Feldman (BIU)
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Place: Seminar room

We continue

- , Diamond on Kurepa trees, Ziemek Kostana (BIU)
Ziemek Kostana (BIU)
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Place: Seminar room

We shall prove independence results concerning diamond on Kurepa trees. This is joint work with Assaf Rinot and Saharon Shelah.

- , Forcing with models as side conditions, part 1, Ido Feldman (BIU)
Ido Feldman (BIU)
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Place: Seminar room, second floor

A decade ago, Neeman introduced a new method of forcing using two-type of models as side conditions. In 2015, Kuziljevic and Todor\'cevi\'c generalized Todor\'cevi\'c's   $\epsilon$-chain of elementary sub-models to a matrices of countable elementary sub-models which are used to force a $\diamond^+$ sequence and fast clubs on $\omega_1$. A year ago, Curial Gallart Rodr\'iguez was able to combine this two methods and thus introducing a two-type symmetric matrices.
We will go through the paper of Curial and if time allows present some minor applications.

- , Minimal non-sigma-scattered linear order, part 2, Roy Shalev (BIU)
Roy Shalev (BIU)
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Place: seminar room

Continuing the lecture from June, we show how to furthermore get a large antichain of minimal non-sigma-scattered linear orders

- , Universality problem for wide Aronszajn trees, part 3, Omer Ben-Neria (HUJI)
Omer Ben-Neria (HUJI)
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Place: Seminar room

We continue

- , Universality problem for wide Aronszajn trees, part 2, Omer Ben-Neria (HUJI)
Omer Ben-Neria (HUJI)
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Place: Seminar room

We continue

- , Universality problem for wide Aronszajn trees, part 1, Omer Ben-Neria (HUJI)
Omer Ben-Neria (HUJI)
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Place: Seminar room, math depratment

The university problem for a class K of structures asks if there is a structure M in K such that every other structure in K embeds into. I will start by relating the universality problem for the class of wide Aronszajn trees to generalized descriptive set theory, and specifically to generalizations of Scott's theorem on the complexity of isomorphism classes in generalized Baire spaces. Next, we will see that in L there is a strong failure of this universality problem. Finally, I will describe a consistency result showing that a universal wide Aronszajn tree on omega_2 can exist relative to a weakly compact cardinal.  The new results are from a joint work with Menachem Magidor and Jouko Vaananen.

- , On generlized Ehrenfeucht-Mostowski models, Ido Feldman (BIU)
Ido Feldman (BIU)
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Place: Seminar room

One of the main question in generalized descriptive set theory is whether there is generalized Borel-reducibility counterpart of Shelah's main gap theorem.
Which is stated as follows:
Let $T_1$ be a classifiable theory and $T_2$ a non-classifiable theory. Is there a Borel reduction from the isomorphism relation of $T_1$ to the isomorphism relation of $T_2$?

This question was studied by Friedman, Hyttinen, and Weinstein (former Kulikov), who gave a positive answer to this question for the case $\kappa$ a successor cardinal (under certain cardinal assumptions), and $T_2$ a stable unsuperstable theory.
All of the above assume the existence of diamond on a stationary set in order to execute the proof.
The case when $\kappa$ is strongly-inaccessible cardinal was study by Hyttinen and Moreno.

In a joint paper with Moreno, a positive answer to this question is given for $\kappa$ an inaccessible cardinal, $T_2$ a stable theory with OCP.

In this talk we give a brief overview of the subject and present new result which extends to unsuperstable theories for strongly inaccessible cardinal.
Thus, in particular giving a positive answer to the question without the use of diamond.

- , Restrictions of Ultrapower Embeddings from Generic Extensions to their Ground Model, Eyal Kaplan (TAU)
Eyal Kaplan (TAU)
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Place: Math department, Seminar room
Assume that V[G] is a generic (set) forcing over the ground model V, and let \kappa be a measurable cardinal in V[G].
Let  j_W \colon V[G] \to M  be an ultrapower embedding associated with a normal measure W\in V[G]  on \kappa. 
What can be said about j_W\restriction_{V}? Is it definable in V? Is it an iterated ultrapower of V by its measures/extenders?
We will survey some known results. These results do not apply to the case where the forcing being used is an iteration of Prikry forcings. 
Our main focus will be to present new tools for dealing with iterations of Prikry forcings, taken with various supports (Nonstationary support, Full support and Easton support).
This is a joint work with Moti Gitik.
- , Reflecting topological spaces through elementary submodels, part 3, Rodrigo Rey Carvalho (BIU)
Rodrigo Rey Carvalho (BIU)
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Place: Math department, Seminar room

We continue

- , Reflecting topological spaces through elementary submodels, part 2, Rodrigo Rey Carvalho (BIU)
Rodrigo Rey Carvalho (BIU)
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Place: Math department, Seminar room

We continue

- , Reflecting topological spaces through elementary submodels, part 1, Rodrigo Rey Carvalho (BIU)
Rodrigo Rey Carvalho (BIU)
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Place: Building 504, room 1

In this series of talks I will present the notion of $X_M$, that is, a topological space $X$ as seen through an elementary submodel $M$. This concept first appeared in the paper ``The topology of elementary submodels", by L. Junqueira and F. Tall. One of the central topics of this field is to question whether topological properties can be preserved through this operation, both ``upwards" and ``downwards".  I will focus on covering properties, giving a panorama of where we stand currently, presenting some results I obtained during my Ph.D.

- , Minimal non-sigma-scattered linear order of size an inaccessible, part 1, Roy Shalev (BIU)
Roy Shalev (BIU)
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Place: Building 504, Room 1

In a recent paper, Cummings, Eisworth and Moore gave the first consistent example of a minimal non-sigma-scattered linear order of size greater than Aleph_1. Their construction works for arbitrary successor cardinals, and leaves open the case of inaccessibles. In this talk, we will give a construction that covers this missing case.

- , Weakly reflecting graph properties, Attila Joó (Hamburg)
Attila Joó (Hamburg)
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Place: Online

We'll discuss our recent preprint.

 

- , Infinite game theory, part 3, Ron Peretz (BIU)
Ron Peretz (BIU)
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Place: Building 504, Room 1

We continue

- , Infinite game theory, part 2, Ron Peretz (BIU)
Ron Peretz (BIU)
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Place: Building 504, Room 1

We continue

- , Infinite game theory, part 1, Ron Peretz (BIU)
Ron Peretz (BIU)
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Place: Building 504, Room 1

An invitation to the subject

- , Saturation properties of ideals, part 2, Tanmay Inamdar (BGU)
Tanmay Inamdar (BGU)
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Place: Building 504, Room 1

TBA

- , Saturation properties of ideals, part 1, Tanmay Inamdar (BGU)
Tanmay Inamdar (BGU)
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Place: Building 504, Room 1

TBA

- , Controlling the existence of isomorphisms between Souslin trees, Shira Greenstein-Yadai (BIU)
Shira Greenstein-Yadai (BIU)
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Place: Building 504, Room 1

TBA

- , Club isomorphisms on higher Aronszajn trees, Roy Shalev (BIU)
Roy Shalev (BIU)
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Place: Building 216, Room 201

We will present the result of Krueger that assuming an ineffable cardinal it is consistent that CH holds and for any two normal countably closed $\omega_2$-Aronszajn trees are club-isomorphic.

- , Iterating the construction of inner models from extended logics, part 6, Ur Ya'ar (BIU)
Ur Ya'ar (BIU)
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Place: Building 505, Room 62

The last talk for this series

- , Iterating the construction of inner models from extended logics, part 5, Ur Ya'ar (BIU)
Ur Ya'ar (BIU)
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Place: Building 505, Room 62

We continue

- , Iterating the construction of inner models from extended logics, part 4, Ur Ya'ar (BIU)
Ur Ya'ar (BIU)
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Place: Building 505, Room 62

We continue

- , Iterating the construction of inner models from extended logics, part 3, Ur Ya'ar (BIU)
Ur Ya'ar (BIU)
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Place: Building 505, Room 62

We continue

- , Iterating the construction of inner models from extended logics, part 2, Ur Ya'ar (BIU)
Ur Ya'ar (BIU)
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Place: Building 505, Room 62

We continue

- , Iterating the construction of inner models from extended logics, part 1, Ur Ya'ar (BIU)
Ur Ya'ar (BIU)
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Place: Building 505, Room 62

One way of generalizing Goedel's constructible universe L, is to replace the notion of definability used at successor stage, and take all subsets of the last stage which are definable using a logic L* extending first order logic. This will result in a model of ZF, denoted C(L*). In some cases it will also be a model of AC. As in the case of L, we can formulate the axiom "V=C(L*)", but unlike L, it is not always the case that C(L*) |= "V=C(L*)", that is, when we construct C(L*) inside C(L*) we might get a smaller model. If that is the case, we can iterate this construction, and if it doesn't stabilize, continue transfinitely with intersections at limit stages.
In this series of lectures we'll present some of the results concerning these sequences of iterated constructible models, focusing mainly on the case of the logic obtained by adding the cofinality-omega quantifier, and the case of Stationary logic. We'll show that in the first case the possibilities are rather limited, and that in the second case we can get almost everything we want.

- , Sealing Kurepa Trees in ZFC, Itamar Giron (HUJI)
Itamar Giron (HUJI)
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Place: Building 505, room 62

A class of trees may be defined using shared properties: height, width, cardinality of  the branch set, etc. For a tree T within a model M we ask if we may add a branch (or branches) to the model using some forcing notion without destroying said properties. If we cannot, we say the tree is sealed. A natural followup question is, under what conditions can we build a forcing notion which does not harm the properties of the tree, and seals it? In this lecture I show that given a Kurepa tree in M without any other assumptions on the model (except that it is a model of ZFC), we can construct a proper forcing-notion which forces a tree to be sealed for every forcing-notion from the ground model, without collapsing \aleph_2 and \aleph_1 .

- , Partitions on topological spaces and a club-like principle, part 4, Rodrigo Rey Carvalho (BIU)
Rodrigo Rey Carvalho (BIU)
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Place: Building 505, room 62

We continue

- , Partitions on topological spaces and a club-like principle, part 3, Rodrigo Rey Carvalho (BIU)
Rodrigo Rey Carvalho (BIU)
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Place: Building 505, room 62

We continue

- , Partitions on topological spaces and a club-like principle, part 2, Rodrigo Rey Carvalho (BIU)
Rodrigo Rey Carvalho (BIU)
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Place: Building 505, Room 62

We continue

- , Partitions on topological spaces and a club-like principle, part 1, Rodrigo Rey Carvalho (BIU)
Rodrigo Rey Carvalho (BIU)
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Place: Building 505, Room 62

In this talk we explore some results from P. Komjáth and W. Weiss from the paper "Partitioning topological spaces in countably many pieces". First we address a problem found on a proof of a theorem, regarding the Cantor-Bendixson decomposition. Then we revisit an example from this paper made with $\diamondsuit$ and present a new example with the same purpose, but without the use of CH. For this we introduce the new principle $\clubsuit_{F}$. This talk will cover the contents of a joint work with G. Fernandes and L. Junqueira

- , The connection between ladder system uniformization and the Whitehead problem, part 4, Márk Poór (HUJI)
Márk Poór (HUJI)
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Place: Seminar room

We continue

- , The connection between ladder system uniformization and the Whitehead problem, part 3, Márk Poór (HUJI)
Márk Poór (HUJI)
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Place: Seminar room

We continue

- , The connection between ladder system uniformization and the Whitehead problem, part 2, Márk Poór (HUJI)
Márk Poór (HUJI)
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Place: Seminar room

We continue

- , The connection between ladder system uniformization and the Whitehead problem, part 1, Márk Poór (HUJI)
Márk Poór (HUJI)
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Place: Seminar room

We give a brief overview on what had been already known, and present our new result asserting that (appropriate) uniformization principles grant not only the existence of almost free but non-free Whitehead groups, but imply that each strongly $\aleph_1$ abelian group of power $\aleph_1$ has the Whitehead property. This is joint work with S. Shelah (paper #486 in Shelah's list).

- , A minimal Magidor-type forcing (countable case), Zhixing You (BIU)
Zhixing You (BIU)
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Place: seminar room

In their paper from 2013, Koepke, Rasch and Schlicht defined a minimal Prikry-type forcing, which satisfies that any intermediate model is either the ground model or the generic extension.

In this talk, we try to generalize this result, and prove that for a countable limit ordinal delta, we can define a minimal Magidor-type forcing, which adds an increasing continuous sequence C_G of length delta such that any intermediate model between the ground model and the generic extension is of the form "V[C_G restricted to alpha]" for some limit ordinal alpha <= delta.

- , On the delta-strongly compact cardinal, part 2, Zhixing You (BIU)
Zhixing You (BIU)
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Place: Seminar room

We continue

- , On the delta-strongly compact cardinal, part 1, Zhixing You (BIU)
Zhixing You (BIU)
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Place: TBD

Bagaria and Magidor introduced a weak version of strong compactness, δ-strong compactness, which characterizes some compactness properties successfully.
Besides, they also find that the least δ-strongly compact cardinal has odd properties. For example, the least δ-strongly compact cardinal may be singular, which separates δ-strong compactness from strong compactness.

In this series of talk, we will explore some basic properties of δ-strongly compact cardinals, and identity crisis phenomenon among δ-strongly compact cardinals for different δ.

This is joint work with Jiachen Yuan.
 

- , Collapsing successors of singular cardinals using a theorem of Magidor, Inber Oren (BGU)
Inber Oren (BGU)
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Place: Seminar room

In their paper [1] Adolf, Apter and Koepke showed that by assuming enough super-compactness for kappa one can prove that Aleph_{w1 + 1} can be collapsed to have cofinality w1 while preserving Aleph_{w1} and all cardinals above Aleph_{w1 + 1}. For this, they use Magidor's work [2] to get the required two-stage forcing.

The main goal of the lecture is to address the following natural question: When does this forcing preserve cardinals below kappa? We shall prove that no new bounded subsets of kappa are introduced by the forcing if and only if the cofinality of kappa is countable.

References
[1] Adolf, Apter, and Koepke (2017). Singularizing successor cardinals by forcing. Proc. Amer. Math. Soc. 146 (2018), 773-783.
[2] Magidor, M. (1977). On the singular cardinals problem i. Israel Journal of Mathematics.

- , Forcing with matrices of countable elementary submodels, Roy Shalev (BIU)
Roy Shalev (BIU)
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Place: Seminar room

We present a forcing poset by Kuzeljevic and Todorcevic {https://www.ams.org/journals/proc/2017-145-05/S0002-9939-2017-13133-5/home.html}.
The conditions are finite matrices whose rows  consist of isomorphic countable elementary submodels of a given structure of the form H_theta.
We will show that this forcing poset adds a Kurepa tree.
Moreover, forcing with a "continuous" modification adds an almost Souslin Kurepa tree, i.e., the level set of any antichain in the tree  is not stationary in omega_1.

- , Weak Kurepa trees, slender tree properties, and guessing models, part 3, Sarka Stejskalova (BIU)
Sarka Stejskalova (BIU)
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Place: Building 604, Room 13

we continue

- , Weak Kurepa trees, slender tree properties, and guessing models, part 2, Sarka Stejskalova (BIU)
Sarka Stejskalova (BIU)
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Place: seminar room

we continue

- , Weak Kurepa trees, slender tree properties, and guessing models, part 1, Sarka Stejskalova (BIU)
Sarka Stejskalova (BIU)
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Place: seminar room

The weak Kurepa hypothesis at omega_1 says that there exists a tree of size and height omega_1 which has at least omega_2 many cofinal branches. In the talk we will focus on the negation of the weak Kurepa hypothesis and the connection to the slender tree and ineffable slender tree properties at omega_2.

In the first part of the talk(s) we will focus on the negation of the weak Kurepa hypothesis and its effect on cardinal arithmetic. In the second part of the talk(s) we define the slender tree property and show that the slender tree property implies the negation of weak Kurepa hypothesis. In the last part of the talk(s) we will define the guessing model principle at omega_2 and its variants and compare them with the slender tree properties.

- , An Aronszajn line with no countryman suborders, Roy Shalev (BIU)
Roy Shalev (BIU)
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Place: Seminar room

Moore proved it is consistent assuming the existence of a supercompact cardinal that the class of uncountable linear orders has a five element basis.
The elements are X, w1, the dual of w1, C, and the dual of C, where X is any suborder of the reals of size w1, and C is any Countryman line.
This raises the question of the existence of an Aronszajn line with no countryman suborder, we will present such a construction by Moore from the combinatorial principle Mho.

- , From Sierpinski-type colourings to Ulam-type matrices, Tanmay Inamdar (BIU)
Tanmay Inamdar (BIU)
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Place: zoom

Ulam matrices were introduced by Ulam in his study of the measure problem. Ulam’s construction applies to all successor cardinals Kappa, and later Hajnal extended the construction to apply to some limit cardinals as well. In my talk I will show how a colouring principle introduced by Sierpinski can be used to construct matrices with similar applications as the matrices of Ulam and Hajnal. I will also show how such colouring principles can be obtained from the existence of a non-trivial C-sequence on Kappa using walks on ordinals. As a consequence, the resulting matrices are more readily available than the matrices of Ulam and Hajnal.

The results I present are joint work with Assaf Rinot.

- , A Galvin-Hajnal theorem for generalized cardinal characteristics, part 2, Chris Lambie-Hanson (Czech Academy of Sciences)
Chris Lambie-Hanson (Czech Academy of Sciences)
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Place: Seminar room

we continue

- , A Galvin-Hajnal theorem for generalized cardinal characteristics, part 1, Chris Lambie-Hanson (Czech Academy of Sciences)
Chris Lambie-Hanson (Czech Academy of Sciences)
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Place: Seminar room

We prove that a variety of generalized cardinal characteristics,  including meeting numbers, the reaping number, and the dominating number, satisfy an analogue of the Galvin-Hajnal theorem, and hence also of Silver's theorem, at singular cardinals of uncountable cofinality. In the first talk, we will introduce some fundamental PCF-theoretic facts due to Shelah and Jech that will be used in our proof, and in the second talk we will prove our generalized Galvin-Hajnal theorem.

- , Ramsey degrees and topological partition theorems, Jing Zhang (BIU)
Jing Zhang (BIU)
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Place: Building 305, Room 133

We will go over the theorem of Todorcevic and Raghavan that any finite coloring on pairs from an aleph1 subset of the reals, we can find an almost monochromatic subset homeomorphic to the rationals.

- , Galvin's Problem In Higher Dimensions, Ido Feldman (BIU)
Ido Feldman (BIU)
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Place: Building 305, Room 133

In 1970's Galvin conjectured that for all colorings of the reals there is a subset H of the reals homeomorphic to the rational numbers that gets at most two colors. i.e. the 2-dimensional Ramsey degree of the rational with respect to the reals is 2. Baumgartner proved that if we consider any infinite countable Hausdorff space X then there is a coloring with \omega many colors, which takes all its values on subsets homeomorphic to \mathbb{Q}. In paper from 2018 Raghavan and Todorcevic showed that assuming existance of Woodin cardinal the 2-dimensional Ramsey degree of \mathbb{Q} within respect to the class of all uncountable sets is 2. In this talk we give a result of Raghavan and Todorcevic in higher dimensions which is that there is no generalization of their previous theorem to dimension 3 and higher.

- , Construction of id_p(C,I)-positive coloring, Ido Feldman (BIU)
Ido Feldman (BIU)
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Place: Building 305, Room 133

For kappa Mahlo and a stationary subset of kappa that does not reflect at regulars, we follow Hoffman's proof to show that for Shelah's ideal id_p(C,I), there is a coloring c:[\kappa]^2\rightarrow\kappa such that, for every unbounded set A, c[[A]^2] is measure one with respect to id_p(C,I)

- , On weak and middle diamonds, part 2, Tanmay Inamdar (BIU)
Tanmay Inamdar (BIU)
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Place: Building 305, Room 133

we continue

- , On weak and middle diamonds, part 1, Tanmay Inamdar (BIU)
Tanmay Inamdar (BIU)
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Place: Building 305, Room 133

We'll survey some classical works on weak diamond and more recent works on middle diamond

- , Nearly special free Souslin tree, Shira Greenstein (BIU)
Shira Greenstein (BIU)
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Place: TBA

For an infinite cardinal lambda and a cardinal chi > 1 such that lambda^{<chi}=lambda, assuming an instance of the proxy principle at lambda^+, we construct a chi-free lambda^+-Souslin tree whose chi power is special.

- , Combinatorial number theory with nonstandard methods, Jing Zhang (BIU)
Jing Zhang (BIU)
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Place: Building 305, Room 133

We will survey applications of nonstandard analysis to prove theorem in additive Ramsey theory and combinatorial number theory. We will use Jin's sumset theorem and the recent solution of the Erdos' sumset conjecture as illustrating examples.

- , Cofinal types on w1 and w2, part 10, Roy Shalev (BIU)
Roy Shalev (BIU)
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Place: Building 216, Room 201

we continue

- , Cofinal types on w1 and w2, part 9, Roy Shalev (BIU)
Roy Shalev (BIU)
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Place: Building 216, Room 201

we continue

- , Colouring orders and ordering trees, Miguel Moreno (Vienna)
Miguel Moreno (Vienna)
-
Place: zoom

One of the main motivations of generalised descriptive set theory is to use the Borel reducibility to study the complexity of theories. Following that motivation, one of the main question is whether the isomorphism relation of any classifiable theory is Borel reducible to the isomorphism relation of any non-classifiable theory. This has been proved to be consistent and under certain cardinality assumptions, the isomorphism relation of any classifiable theory is Borel reducible to the isomorphism relation of any stable unsuperstable theory.
In this talk we will study the isomorphism relation of unstable theories by merging Shelah's ordered trees method to construct models of unsuperstable theories and Hyttinen-Kulikov's coloured tree to construct Borel reductions.

- , When Luzin met Souslin, part 2, Assaf Rinot (BIU)
Assaf Rinot (BIU)
-
Place: Building 216, Room 201

We continue with the derivation of club_AD from a Luzin set, and then point out how to also get the same conclusion from the stick principle.

- , When Luzin met Souslin, part 1, Assaf Rinot (BIU)
Assaf Rinot (BIU)
-
Place: zoom

In a previous joint work with Shalev, we introduced the guessing principle club_AD, proved that it follows from the existence of a Souslin tree, and that it is sufficient for the construction of a Dowker S-space, as well as an O-space. Here we show that the same principle follows from the existence of a Luzin set.

This is joint work with Roy Shalev.

- , Forcing-theoretic approach to universal homogeneous structures, part 5, Ziemek Kostana (BIU)
Ziemek Kostana (BIU)
-
Place: Area 502, Room 28

This is the last lecture for this series

- , Forcing-theoretic approach to universal homogeneous structures, part 4, Ziemek Kostana (BIU)
Ziemek Kostana (BIU)
-
Place: Building 504, Room 2

We continue

- , Forcing-theoretic approach to universal homogeneous structures, part 3, Ziemek Kostana (BIU)
Ziemek Kostana (BIU)
-
Place: Building 504, Room 2

We continue

- , Forcing-theoretic approach to universal homogeneous structures, part 2, Ziemek Kostana (BIU)
Ziemek Kostana (BIU)
-
Place: Building 504, Room 2

We continue

- , Forcing-theoretic approach to universal homogeneous structures, part 1, Ziemek Kostana (BIU)
Ziemek Kostana (BIU)
-
Place: Building 504, Room 2

In this series of talks, we will investigate relations between generic first-order structures added by forcing, and universal homogeneous structures arising from the Fraisse theory. During the first talk, I intend to give a self-contained introduction to the Fraisse theory, define forcings Fn(\kappa, K, \lambda), and explain why are they provide a reasonable generalization of Fraisse theory to higher cardinalities.

- , Pairwise far Souslin trees, Shira Greenstein (BIU)
Shira Greenstein (BIU)
-
Place: Building 504, Room 2

We will present a construction of a family of 2^kappa many kappa-Souslin trees for which the product of any finitely many of them is still kappa-Souslin. The proof uses the Brodsky-Rinot proxy principle and hence applies to any regular uncountable cardinal kappa.

- , Cofinal types on w1 and w2, part 8, Roy Shalev (BIU)
Roy Shalev (BIU)
-
Place: Building 504, Room 2

We continue

- , Cofinal types on w1 and w2, part 7, Roy Shalev (BIU)
Roy Shalev (BIU)
-
Place: Building 504, Room 2

We continue

- , Cofinal types on w1 and w2, part 6, Roy Shalev (BIU)
Roy Shalev (BIU)
-
Place: Building 216, Room 132

We continue

- , Cofinal types on w1 and w2, part 5, Roy Shalev (BIU)
Roy Shalev (BIU)
-
Place: seminar room

We continue

- , Cofinal types on w1 and w2, part 4, Roy Shalev (BIU)
Roy Shalev (BIU)
-
Place: Seminar room

We continue

- , Cofinal types on w1 and w2, part 3, Roy Shalev (BIU)
Roy Shalev (BIU)
-
Place: Seminar room

we continue

- , Cofinal types on w1 and w2, part 2, Roy Shalev (BIU)
Roy Shalev (BIU)
-
Place: Seminar room

we continue

- , Cofinal types on w1 and w2, part 1, Roy Shalev (BIU)
Roy Shalev (BIU)
-
Place: seminar room

This is a series of talks motivated by a recent paper by Kuzeljevic and Todorcevic

- , Higher-dimensional Delta-systems, Ari Brodsky (SCE)
Ari Brodsky (SCE)
-
Place: seminar room

We present a result of Chris Lambie-Hanson, investigating higher-dimensional Δ-systems, isolating a particular definition thereof and proving a higher-dimensional version of the classical Δ-system lemma.
https://arxiv.org/abs/2006.01086

- , The dominating number at singular strong limits, part 3, Tanmay Inamdar (BIU)
Tanmay Inamdar (BIU)
-
Place: seminar room

we continue

- , Incompactness of the second uncountable cardinal, Assaf Rinot (BIU)
Assaf Rinot (BIU)
-
Place: Seminar room

In a celebrated paper from 1997, Shelah proved that Pr1(w2,w2,w2,w) is a theorem of ZFC, and it remains open ever since whether moreover Pr1(w2,w2,w2,w1) holds.

In an unpublished note from 2017, Todorcevic proved that a certain weakening of the latter follows from CH.
In a recent paper with Zhang (arXiv:2104.15031), we gave a few weak sufficient conditions for Pr1(w2,w2,w2,w1) to hold.
In an even more recent paper (arXiv:1910.02419v2), Shelah proved that it holds, assuming the existence of a nonreflecting stationary subset of S^2_0, hence, the strength of the failure is at least that of a Mahlo cardinal.
In this talk, we'll prove that (weak forms of) square(w2) are sufficient for Pr1(w2,w2,w2,w1) to hold, thus raising the strength of the failure to that of a weakly compact cardinal.

This is joint work with Jing Zhang.
 

- , The dominating number at singular strong limits, part 2, Tanmay Inamdar (BIU)
Tanmay Inamdar (BIU)
-
Place: Seminar room

we continue

- , The dominating number at singular strong limits, part 1, Tanmay Inamdar (BIU)
Tanmay Inamdar (BIU)
-
Place: seminar room

I will talk about the principle SEP of Shelah. As an application, I will prove a result from Shelah 1159, that the dominating number always takes the maximal value at a strong limit singular cardinal.

- , Coloring cubes, part 3, Ido Feldman (BIU)
Ido Feldman (BIU)
-
Place: Building 105, Room 101

last talk for this series

- , Coloring cubes, part 2, Ido Feldman (BIU)
Ido Feldman (BIU)
-
Place: seminar room

we continue

- , Coloring cubes, part 1, Ido Feldman (BIU)
Ido Feldman (BIU)
-
Place: seminar room

We discuss colorings of triples

- , Kurepa trees and ccc forcing, Tanmay Inamdar (BIU)
Tanmay Inamdar (BIU)
-
Place: seminar room
- , Kurepa trees, part 4, Roy Shalev (BIU)
Roy Shalev (BIU)
-
Place: seminar room

we continue

- , Kurepa trees, part 3, Roy Shalev (BIU)
Roy Shalev (BIU)
-
Place: seminar room

we continue

- , Kurepa trees, part 2, Roy Shalev (BIU)
Roy Shalev (BIU)
-
Place: seminar room

we continue

- , Strong Prikry Property for Magidor Iteraton, part 3, Omer Ben-Neria (HUJI)
Omer Ben-Neria (HUJI)
-
Place: zoom

link to recording

- , Kurepa trees, part 1, Roy Shalev (BIU)
Roy Shalev (BIU)
-
Place: seminar room

We discuss consistency results concerning Kurepa trees

- , Strong Prikry Property for Magidor Iteraton, part 2, Omer Ben-Neria (HUJI)
Omer Ben-Neria (HUJI)
-
Place: zoom

In his celebrated work on the identity crisis of strongly compact cardinals, Magidor introduced a special iteration of Prikry forcings for a set of measurable cardinals, known as the Magidor iteration.

The purpose of this talk is to state and prove a version of the strong Prikry Lemma for such iterations, extending a result of Fuchs for the case where the set of measurables is discrete.  We will also describe several applications regarding the genericity of sequences of critical points in iterated ultrapowers.

Link to recording.

- , The strong Prikry Property for Magidor iterations, Omer Ben-Neria (HUJI)
Omer Ben-Neria (HUJI)
-
Place: zoom

In his celebrated work on the identity crisis of strongly compact cardinals, Magidor introduced a special iteration of Prikry forcings for a set of measurable cardinals, known as the Magidor iteration.

The purpose of this talk is to state and prove a version of the strong Prikry Lemma for such iterations, extending a result of Fuchs for the case where the set of measurables is discrete.  We will also describe several applications regarding the genericity of sequences of critical points in iterated ultrapowers.
 
Link to recording.
- , Stationary reflection and Prikry forcing, part 2, Yair Hayut (HUJI)
Yair Hayut (HUJI)
-
Place: zoom

we continue

- , Stationary reflection and Prikry forcing, part 1, Yair Hayut (HUJI)
Yair Hayut (HUJI)
-
Place: Zoom
 
In 1982, Magidor proved the consistency of stationary reflection at \aleph_{\omega+1}, relative to an \omega-sequence of supercompact cardinals. 
Square based heuristics indicated that a much weaker large cardinal hypothesis is the correct strength. 
In a sequence of results of various authors, Magidor's result was gradually improved to stationary reflection at all sets except one "bad" stationary set at \aleph_{\omega+1}, starting with a large cardinal property weaker than \kappa^+-supercompactness. 
In a joint work with Unger, we managed to obtain the consistency of (full) stationary reflection, from what seems to be close to the optimal hypothesis.
In this talk I will present the main ideas behind the proof (which is the interplay between Prikry type forcings and iterated ultrapowers). This method shares some features with the Sigma-Prikry framework, where the main difference is its non-iterative nature.
In a joint work with Ben-Neria, we tackled the problem of combining the failure of SCH with stationary reflection, starting with a similar large cardinal hypothesis. 

In order to do that, we used a similar analysis of the extender based Prikry forcing.

 

Link to video recording.

- , All colorings are strong, but some colorings are stronger than the others, Assaf Rinot (BIU)
Assaf Rinot (BIU)
-
Place: zoom

Strong colorings are everywhere - they can be obtained from analysis of basis problems, transfinite diagonalizations, oscillations, or walks on ordinals. They give rise to interesting topological spaces and partial orders.

In this talk, I'll be looking at all aspects mentioned above, reporting on findings from my joint projects with Kojman, Lambie-Hanson, Inamdar, Steprans and Zhang.
- , The spectra of cardinalities of branches of Kurepa trees, part 2, Mark Poor (HUJI)
Mark Poor (HUJI)
-
Place: zoom

we continue

- , The spectra of cardinalities of branches of Kurepa trees, part 1, Mark Poor (HUJI)
Mark Poor (HUJI)
-
Place: zoom
We study the possible values of the Kurepa spectra, i.e. how the set of cardinalities of branches of Kurepa trees may look like. It turns out that assuming GCH below the second uncountable cardinal (and possibly the existence of an inaccessible) we can force every set of cardinals (satisfying some obvious necessary conditions) to be the Kurepa spectrum.
- , Fresh subsets of measurable ultrapowers, Philip Luecke (Barcelona)
Philip Luecke (Barcelona)
-
Place: zoom
In my talk, I want to present recent results studying the closure and non-closure properties of measurable ultrapowers with respect to Hamkin's notion of freshness. These results show that the extent of these properties highly depends on the combinatorics of the underlying model of set theory. While a result of Sakai shows that it is possible to obtain ultrapowers with maximal closure properties by forcing over a model containing a strongly com- pact cardinal, it turns out that measurable ultrapowers of canonical inner models possess the minimal amount of closure properties. The proof of this result heavily makes use of the existence of various square sequences in these models. This is joint work with Sandra Muller (Vienna).
- , Universal functions, strong colourings and ideas from PID, Juris Stperans (York)
Juris Stperans (York)
-
Place: zoom

A construction of Shelah will be reformulated using the PID to provide alternative models of the failure of CH and the existence of a universal colouring of cardinality . The impact of the range of the colourings will be examined. An application to the theory of strong colourings over partitions will also be given.

Links to the recording.

- , Actions of tame abelian product groups, Assaf Shani (Harvard)
Assaf Shani (Harvard)
-
Place: zoom

A Polish group G is tame if for any continuous action of G, the corresponding orbit equivalence relation is Borel. Suppose that G=\prod_n \Gamma_n is a product of countable abelian groups. It follows from results of Solecki and Ding-Gao that if G is tame, then all of its actions are in fact potentially \Pi^0_6. Ding and Gao conjectured that this bound could be improved to \Pi^0_3. We refute this, by finding an action of a tame abelian product group, which is not potentially \Pi^0_5.
The proof involves forcing over models where the axiom of choice fails for sequences of finite sets.
This is joint work with Shaun Allison.

- , S spaces and L spaces, part 2, Roy Shalev (BIU)
Roy Shalev (BIU)
-
Place: zoom
We introduce a new combinatorial principle which we call ♣_AD. This principle asserts the existence of a certain multi-ladder system with guessing and almost-disjointness features, and is shown to be sufficient for carrying out de Caux type constructions of topological spaces.
Our main result states that strong instances of ♣_AD follow from the existence of a Souslin tree.  As an application, we obtain a simple, de Caux type proof of Rudin’s result that if there is a Souslin tree, then there is an S-space which is Dowker.

 

- , S spaces and L spaces, part 1, Roy Shalev (BIU)
Roy Shalev (BIU)
-
Place: zoom

An S-space is a regular topological space that is hereditarily separable but not Lindel\"of. An L-space is a regular topological space that is hereditarily Lindel\"of but not separable. We will survey the history behind the question of their existence and present some constructions

- , Woodin Extender Algebra and its applications for absoluteness, part 2, Menachem Magidor (HUJI)
Menachem Magidor (HUJI)
-
Place: zoom

we continue

- , Woodin Extender Algebra and its applications for absoluteness, part 1, Menachem Magidor (HUJI)
Menachem Magidor (HUJI)
-
Place: Zoom

This talk will survey known results and will be the first of several talks which will not necessarily follow in the consecutive weeks.                  

- , Independent families in the countable and the uncountable, Vera Fischer (Vienna)
Vera Fischer (Vienna)
-
Place: Zoom

Independent families on $\omega$ are families of infinite sets of integers with the property that for any two finite subfamilies $A$ and $B$ the set $\bigcap A\backslash \bigcup B$ is infinite. Of particular interest are the sets of the possible cardinalities of maximal independent families, which we refer to as the spectrum of independence. Even though we do have the tools to control the spectrum of independence at $\omega$ (at least to a  large extent), there are many relevant questions regarding higher counterparts of independence in generalised Baire spaces, which remain widely open.

- , Higher Chang Conjecture, part 2, Yait Hayut (HUJI)
Yait Hayut (HUJI)
-
Place: Zoom

Here is the recording

- , Higher Chang Conjecture, Yait Hayut (HUJI)
Yait Hayut (HUJI)
-
Place: zoom
In this talk I will present some results regarding the consistency strength of Higher variants of Chang's Conjecture. 
I will start with the classical result by Silver of Chang's Conjecture from $\omega_1$-Erdos cardinal.  
Then, I will give an upper bound for the consistency strength of  $(\aleph_{\omega+1}, \aleph_{\omega}) -->>(\aleph_1, \aleph_0)$
and $(\aleph_4, \aleph_3) -->> (\aleph_2, \aleph_1)$ (joint with Eskew)
from supercompactness assumptions. 
If time permits, I will describe the strategy for obtaining a global result:
(\kappa^+,\kappa) -->> (\mu^+, \mu)
for all regular $\kappa$, and $\mu < \kappa$, and talk about the barriers that we face when trying to extend this result.
 
Link to recording.
- , On Continuous Tree-Like Scales and related properties of Internally Approachable structures, Omer Ben-Neria (HUJI)
Omer Ben-Neria (HUJI)
-
Place: zoom
In his PhD thesis, Luis Pereira isolated and developed several principles of singular cardinals that emerge from Shelah's PCF theory; principles which involve properties of scales, such as the inexistence of continuous Tree-Like scales, and properties of internally approachable structures such as the Approachable Free Subset Property. 
 
In the talk, we will discuss these principles and their relations, and present new results from a joint work with Dominik Adolf concerning their consistency and consistency strength.
 
Link to recording.
- , Local club condensation in extender models, Gabriel Fernandes (BIU)
Gabriel Fernandes (BIU)
-
Place: zoom

Local club condensation is an abstraction of the condensation properties of the constructible hierarchy.
We will prove that for extender models that are countably iterable, given a cardinal kappa, the J_alpha^{E} hierarchy witnesses local club condensation in the interval (kappa^+,kappa^++) if and only if kappa is not a subcompact cardinal in L[E]. 

From the above and the equivalence between subcompact cardinals and square, due to Schimmerling and Zeman, it follows that in such extender models \square_kappa holds iff the J_alpha^{E} hierarchy witnesses that local club condensation holds in the interval (kappa^+,kappa^++).

- , Coloring Superstable Graphs, Yatir Halevi (BGU)
Yatir Halevi (BGU)
-
Place: zoom

Given a graph G=(V,E), a coloring of G in \kappa colors is a map c:V\to \kappa in which adjacent vertices are colored in different colors. The chromatic number of G is the smallest such \kappa.

We will briefly review some questions and conjectures on the chromatic number of infinite graphs and will mainly concentrate on the following strong form of Taylor's conjecture:
If G is an infinite graph with chromatic number\geq \alepha_1 then it contains all finite subgraphs of Sh_n(\omega) for some n, where Sh_n(\omega) is the n-shift graph (which we will introduce).

 
The conjecture was disproved by Hajnal-Komjath. However, we will present a proof for \omega-stable graphs and a proof of a generalization for superstable graphs. If time permits will discuss stable graphs in general.
No previous knowledge in model theory or graph theory is needed.

Joint work with Itay Kaplan and Saharon Shelah.

- , On the inexistence of Continuous Tree-Like Scales, and the Approachable Free Subset Property, Omer Ben-Neria (HUJI)
Omer Ben-Neria (HUJI)
-
Place: Building 216, Room 132

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.

- , Jensen's covering theorem for L, Gabriel Fernandes (BIU)
Gabriel Fernandes (BIU)
-
Place: Building 216, Room 132

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.
- , Weak diamond, Uniformization and its connection to Whitehead's problem, Menachem Magidor (HUJI)
Menachem Magidor (HUJI)
-
Place: zoom

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.

- , Square of Menger groups, Jialiang He (BIU)
Jialiang He (BIU)
-
Place: Building 216, Room 132
 

 

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.
- , Coding well ordering of the reals with ladders, part 5, Uri Abraham (BGU)
Uri Abraham (BGU)
-
Place: zoom

we continue

- , Fedorchuck's space, part 2, Roy Shalev (BIU)
Roy Shalev (BIU)
-
Place: seminar room

we continue

- , Coding well ordering of the reals with ladders, part 4, Uri Abraham (BGU)
Uri Abraham (BGU)
-
Place: ZOOM

we continue

- , Chang's conjecture, part 2, Jing Zhang (BIU)
Jing Zhang (BIU)
-
Place: zoom

we continue

- , Coding well ordering of the reals with ladders, part 3, Uri Abraham (BGU)
Uri Abraham (BGU)
-
Place: zoom

we continue

- , Chang's conjecture, part 1, Jing Zhang (BIU)
Jing Zhang (BIU)
-
Place: zoom

We'll talk about colorings of triples of w2

- , Fedorchuck's space, part 1, Roy Shalev
Roy Shalev
-
Place: seminar room

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.

- , Coding well ordering of the reals with ladders, part 2, Uri Abraham (BGU)
Uri Abraham (BGU)
-
Place: zoom

we continue

- , A Partition Theorem for Scattered Order Types, part 2, Tanmay Inamdar (BIU)
Tanmay Inamdar (BIU)
-
Place: Seminar room, second floor

we continue

- , Coding well ordering of the reals with ladders, part 1, Uri Abraham (BGU)
Uri Abraham (BGU)
-
Place: zoom

Results from the 2002 paper "Coding with Ladders a Well Ordering of the Reals" by Abraham and Shelah.

Recoding is now available.

- , A Partition Theorem for Scattered Order Types, part 1, Tanmay Inamdar (BIU)
Tanmay Inamdar (BIU)
-
Place: Seminar room, second floor

I will talk about the titular paper by Komjath and Shelah. It is a short paper.

- , The tree property at Aleph_{w+1}, Tzoor Plotnikov (HUJI)
Tzoor Plotnikov (HUJI)
-
Place: zoom

Covering Neeman's proof

- , On the free subset number of a topological space and their G_\delta modification, Istvan Juhasz (Renyi Institute)
Istvan Juhasz (Renyi Institute)
-
Place: zoom
- , Wide Aronszajn trees, Mirna Dzamonja (UEA)
Mirna Dzamonja (UEA)
-
Place: zoom

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. 

- , Tameness in Set Theory, Matteo Viale (Torino)
Matteo Viale (Torino)
-
Place: zoom

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

 

- , BPFA and Delta_1-definablity of NS_{w1}, Liuzhen Wu (Chinese Acad. Sciences, Beijing)
Liuzhen Wu (Chinese Acad. Sciences, Beijing)
-
Place: zoom

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.

- , Filter Reflection, Miguel Moreno (KGRC)
Miguel Moreno (KGRC)
-
Place: zoom

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.

- , Compactness problems for chromatic numbers of graphs, Menachem Magidor (HUJI)
Menachem Magidor (HUJI)
-
Place: zoom
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.
 
- , Sigma-Prikry forcing, part 2, Alejandro Poveda (Universitat de Barcelona)
Alejandro Poveda (Universitat de Barcelona)
-
Place: zoom

(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.

- , Sigma-Prikry forcing, part 1, Alejandro Poveda (Universitat de Barcelona)
Alejandro Poveda (Universitat de Barcelona)
-
Place: zoom

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.

- , Transformations of the transfinite plane, Jing Zhang (BIU)
Jing Zhang (BIU)
-
Place: TBA

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.

- , Borel determinacy can not be proved in Zermelo Set Theoy, Menachem Magidor (HUJI)
Menachem Magidor (HUJI)
-
Place: zoom

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.

- , Strong colorings over partitions, Menachem Kojman (BGU)
Menachem Kojman (BGU)
-
Place: Zoom meeting

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.

- , On a paper of Peng and Wu, part 3, Tanmay Inamdar (BIU)
Tanmay Inamdar (BIU)
-
Place: seminar room

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.

- , On a paper of Peng and Wu, part 2, Tanmay Inamdar (BIU)
Tanmay Inamdar (BIU)
-
Place: seminar room

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.

- , On a paper of Peng and Wu, part 1, Tanmay Inamdar (BIU)
Tanmay Inamdar (BIU)
-
Place: Building 604, Room 105

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. 

- , On Shelah 908; or, Strong chains of functions and sets, Tanmay Inamdar (BIU)
Tanmay Inamdar (BIU)
-
Place: seminar room

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). 

- , Indestructibility of some compactness principles, part 2, Šárka Stejskalová (Institute of Mathematics, Prague)
Šárka Stejskalová (Institute of Mathematics, Prague)
-
Place: Building 505, Room 63

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.

- , Indestructibility of some compactness principles, part 1, Šárka Stejskalová (Institute of Mathematics, Prague)
Šárka Stejskalová (Institute of Mathematics, Prague)
-
Place: Area 502, Room 37

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.

- , Constructing Souslin trees without diamond, Ari Meir Brodsky (BIU)
Ari Meir Brodsky (BIU)
-
Place: Area 502, Room 37
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.
- , Compactness principles, GCH and anti-guessing principles, Jing Zhang (BIU)
Jing Zhang (BIU)
-
Place: Area 502, Room 37
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.
- , A guessing principle from a Souslin tree, with an application to topology, part 3, Roy Shalev (BIU)
Roy Shalev (BIU)
-
Place: Area 502, Room 37

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.

- , Some applications of the coloring number, Peter Komjath (ELTE)
Peter Komjath (ELTE)
-
Place: Building 502, Room 37
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.
- , Embedding C*-algebras into the Calkin algebra, Andrea Vaccaro (BGU)
Andrea Vaccaro (BGU)
-
Place: Area 502, Room 37

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.

- , Strong coding trees and applications to Ramsey theory on infinite graphs, Natasha Dobrinen (Denver)
Natasha Dobrinen (Denver)
-
Place: Area 502, Room 37

Abstract is attached

- , Inaccessible Jónsson cardinals, part 3, Shehzad Ahmed (BIU)
Shehzad Ahmed (BIU)
-
Place: Area 502, Room 37

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.

- , Inaccessible Jónsson cardinals, part 2, Shehzad Ahmed (BIU)
Shehzad Ahmed (BIU)
-
Place: Building 502, Room 37
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.
 
- , Inaccessible Jónsson cardinals, part 1, Shehzad Ahmed (BIU)
Shehzad Ahmed (BIU)
-
Place: Building 502, Room 37

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.

- , On the effect of adding kappa^+ many Cohen subsets to kappa, Gabriel Fernandes (BIU)
Gabriel Fernandes (BIU)
-
Place: Building 502, Room 37

This is joint work with Miguel Moreno and Assaf Rinot.

- , PCF and the covering proof, part 2, Dominik Adolf (BIU)
Dominik Adolf (BIU)
-
Place: Building 403, Room 211

This is a continuation of the lecture from last week.

- , PCF and the covering proof, part 1, Dominik Adolf (BIU)
Dominik Adolf (BIU)
-
Place: Building 403, Room 211

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.

- , Rainbow Ramsey theory at uncountable cardinals, part 2, Jing Zhang (BIU)
Jing Zhang (BIU)
-
Place: seminr room

this is a continuation of our previous talk

- , Rainbow Ramsey theory at uncountable cardinals, part 1, Jing Zhang (BIU)
Jing Zhang (BIU)
-
Place: seminar room

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.

- , A guessing principle from a Souslin tree, with an application to topology, part 2, Roy Shalev (BIU)
Roy Shalev (BIU)
-
Place: seminar room

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.

- , A guessing principle from a Souslin tree, with an application to topology, part 1, Roy Shalev (BIU)
Roy Shalev (BIU)
-
Place: seminar room

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.

- , Tree property at the first and double successors with arbitrary gaps, part 4, Alejandro Poveda (Universitat de Barcelona)
Alejandro Poveda (Universitat de Barcelona)
-
Place: Building 105, Room 61
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.
- , Tree property at the first and double successors with arbitrary gaps, part 3, Alejandro Poveda (Universitat de Barcelona)
Alejandro Poveda (Universitat de Barcelona)
-
Place: Building 105, Room 61
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.
- , Tree property at the first and double successors with arbitrary gaps, part 2, Alejandro Poveda (Universitat de Barcelona)
Alejandro Poveda (Universitat de Barcelona)
-
Place: Building 105, Room 61
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.
- , Tree property at the first and double successors with arbitrary gaps, part 1, Alejandro Poveda (Universitat de Barcelona)
Alejandro Poveda (Universitat de Barcelona)
-
Place: Building 105, Room 61
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.
- , A combinatorial argument of p=t, part 2, Jialiang He (BIU)
Jialiang He (BIU)
-
Place: Building 105, Room 61

We will finish the proof from the last meeting.

- , Diamond^{sharp} from LCC, and applications to GDST, Gabriel Fernandes (BIU)
Gabriel Fernandes (BIU)
-
Place: Building 105, Room 61

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

- , A combinatorial argument of p=t, part 1, Jialiang He (BIU)
Jialiang He (BIU)
-
Place: Building 105, Room 61

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.

- , All about mad families, Asger Törnquist (Københavns Universitet)
Asger Törnquist (Københavns Universitet)
-
Place: Building 105, Room 61
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.
- , Variants of the extender algebra and their applications, Ralf Schindler (Universität Münster)
Ralf Schindler (Universität Münster)
-
Place: Building 105, Room 61

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.

- , The Main Gap in the generalized Borel-reducibility hierarchy, Miguel Moreno (BIU)
Miguel Moreno (BIU)
-
Place: Building 105, Room 61

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.

- , Two constructions of gamma spaces, Jialiang He (BIU)
Jialiang He (BIU)
-
Place: Building 105, Room 61

  

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.

  

- , Projective determinacy for games of length $\omega^2$ and longer, Sandra Müller (KGRC)
Sandra Müller (KGRC)
-
Place: Building 105, Room 61

 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.

- , The strength of very small Jonsson cardinals, Dominik Adolf (BIU)
Dominik Adolf (BIU)
-
Place: seminar room

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.

- , Uniformization properties and graph edge colourings, Daniel T. Soukup (KGRC)
Daniel T. Soukup (KGRC)
-
Place: Building 105, Room 61

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.

- , Selection Principles in Mathematics (an overview), Piotr Szewczak, Cardinal Stefan Wyszyński University in Warsaw, Poland
Piotr Szewczak, Cardinal Stefan Wyszyński University in Warsaw, Poland
-
Place: seminar room
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.
- , An introduction to generalized descriptive set theory, part 4, Miguel Moreno (BIU)
Miguel Moreno (BIU)
-
Place: Building 605, Room 13

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.

- , An introduction to generalized descriptive set theory, part 3, Miguel Moreno (BIU)
Miguel Moreno (BIU)
-
Place: Building 605, Room 13
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.
- , An introduction to generalized descriptive set theory, part 2, Miguel Moreno (BIU)
Miguel Moreno (BIU)
-
Place: Building 605, Room 13

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.

- , On countable unions of countable sets, Asaf Karagila (UEA)
Asaf Karagila (UEA)
-
Place: Building 605, Room 13

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.

- , An introduction to generalized descriptive set theory, part 1, Miguel Moreno (BIU)
Miguel Moreno (BIU)
-
Place: Building 605, Room 13

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.

- , Stationary Reflection from subcompactness, Yair Hayut (HUJI)
Yair Hayut (HUJI)
-
Place: Building 605, Room 13

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.

- , Selection covering properties and Cohen reals, part 2, JiaLiang He (BIU)
JiaLiang He (BIU)
-
Place: Building 605, Room 13
We will address preservation theorems under Cohen forcing for various selection covering properties (Menger, Hurewicz, Rothberger and Gamma property).
- , Selection covering properties and Cohen reals, part 1, JiaLiang He (BIU)
JiaLiang He (BIU)
-
Place: Building 605, Room 13
We will introduce four selection covering properties (Menger, Hurewicz, Rothberger and Gamma property), and consider preserving results on these properties under Cohen forcing. 

 

- , Algebraic Ramsey-theoretic results with small monochromatic sets, David Fernández-Bretón (KGRC)
David Fernández-Bretón (KGRC)
-
Place: Building 605, Room 13
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.

 

- , Chang's Conjecture, club-increasing sequences, and P_max forcing, Part 2, Chris Lambie-Hanson (BIU)
Chris Lambie-Hanson (BIU)
-
Place: Building 507, Room 204
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).

 

- , Chang's Conjecture, club-increasing sequences, and P_max forcing, part 1, Chris Lambie-Hanson (BIU)
Chris Lambie-Hanson (BIU)
-
Place: Building 507, Room 204

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.

- , Cardinal characteristics of omega_1, William Chen (BGU)
William Chen (BGU)
-
Place: Building 507, Room 204
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.
- , Uniqueness triples from the diamond axiom, Ari Brodsky (Ariel University)
Ari Brodsky (Ariel University)
-
Place: Building 507, Room 204

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.

- , Filter compactness and squares, Yair Hayut (TAU)
Yair Hayut (TAU)
-
Place: Building 507, Room 204

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.

- , Knaster and friends, Assaf Rinot
Assaf Rinot
-
Place: Building 507, Room 204

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.

- , On some asymptotic variants of club, part 2, Ashutosh Kumar (HUJI)
Ashutosh Kumar (HUJI)
-
Place: Building 507, Room 204

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

- , On some asymptotic variants of club, part 1, Ashutosh Kumar (HUJI)
Ashutosh Kumar (HUJI)
-
Place: Building 505, Room 65

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

- , Circular orders, ultra-homogeneity and topological groups, Michael Megrelishvili (BIU)
Michael Megrelishvili (BIU)
-
Place: Building 505, Room 65
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.
- , Definable pieces in geometric paradoxes, part 2, Spencer Unger (TAU)
Spencer Unger (TAU)
-
Place: Building 505, Room 65

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.

- , Definable pieces in geometrical paradoxes, part 1, Spencer Unger (TAU)
Spencer Unger (TAU)
-
Place: Building 505, Room 65
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.
- , A small Dowker space, part 2, William Chen (BGU)
William Chen (BGU)
-
Place: Building 505, Room 65

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 small Dowker space, part 1, William Chen (BGU)
William Chen (BGU)
-
Place: Building 505, Room 65

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.

- , Uncountable set of reals with a single condensation point, Eilon Bilinsky (TAU)
Eilon Bilinsky (TAU)
-
Place: building 505, Room 65

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.

- , Squares, ascent paths, and chain conditions, part 2, Chris Lambie-Hanson (BIU)
Chris Lambie-Hanson (BIU)
-
Place: Building 505, Room 65
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.

 

- , Squares, ascent paths, and chain conditions, part 1, Chris Lambie-Hanson (BIU)
Chris Lambie-Hanson (BIU)
-
Place: Building 505, Room 65
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.

 

- , Magidor cardinal and Magidor filters, Yair Hayut (TAU)
Yair Hayut (TAU)
-
Place: Building 505, Room 65

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

- , The tree property at w2, Tom Benhamou (TAU)
Tom Benhamou (TAU)
-
Place: building 604, room 103

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.
 

- , Avoiding Quadruples Using a Scale, Thilo Weinert (BGU)
Thilo Weinert (BGU)
-
Place: Building 604, Room 103

the abstract may be found in here.

, Distributive Aronszajn trees, Ari Brodsky (BIU)
Ari Brodsky (BIU)
Place: Building 604, Room 103

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.

- , On possible restrictions of null and meager ideal, part 2, Ashutosh Kumar (HUJI)
Ashutosh Kumar (HUJI)
-
Place: Building 604, Room 103
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.
- , On possible restrictions of null and meager ideal, part 1, Ashutosh Kumar (HUJI)
Ashutosh Kumar (HUJI)
-
Place: Building 604, Room 103

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.

- , Partitioning pairs of sigma-scattered linear orders, part 2, Thilo Weinert (BGU)
Thilo Weinert (BGU)
-
Place: Building 604, Room 103

This is part 2 of last week's talk.

- , Partitioning pairs of sigma-scattered linear orders, part 1, Thilo Weinert (BGU)
Thilo Weinert (BGU)
-
Place: Building 604, Room 103

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.

- , Partition relations and generalized scattered orders, Chris Lambie-Hanson (BIU)
Chris Lambie-Hanson (BIU)
-
Place: Building 604, Room 103

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.

- , Characterizing sigma-scattered linear orders, part 2, William Chen (BGU)
William Chen (BGU)
-
Place: seminar room

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.

- , Characterizing sigma-scattered linear orders, part 1, William Chen (BGU)
William Chen (BGU)
-
Place: seminar room

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.

- , Partitioning a cardinal into fat stationary sets, part 2, Assaf Rinot
Assaf Rinot
-
Place: seminar room

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

- , Partitioning a cardinal into fat stationary sets, part 1, Assaf Rinot
Assaf Rinot
-
Place: seminar room

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.

 

- , On a Formerly New Order Type, part 3, Thilo Weinert (BGU)
Thilo Weinert (BGU)
-
Place: seminar room

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.

- , On a Formerly New Order Type, part 2, Thilo Weinert (BGU)
Thilo Weinert (BGU)
-
Place: seminar room

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.

- , On a Formerly New Order Type, part 1, Thilo Weinert (BGU)
Thilo Weinert (BGU)
-
Place: Building 216, Room 201

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.

- , Finite subsets of the first uncountable ordinal, part 2, Eran Stein
Eran Stein
-
Place: Building 216, Room 201

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.

- , Finite subsets of the first uncountable ordinal, part 1, Eran Stein
Eran Stein
-
Place: seminar room

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.

- , An anti-Hindman theorem for the real line, Dani Livne
Dani Livne
-
Place: Room 201

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.

- , The colouring number of infinite graphs, part 2, Omri Marcus
Omri Marcus
-
Place: seminar room

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.

- , The colouring number of infinite graphs, part 1, Omri Marcus
Omri Marcus
-
Place: Building 502, Room 9

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.

- , Rigid trees vs. homogeneous trees, Guy Kapon
Guy Kapon
-
Place: Building 502, Room 9

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.

- , The Ostaszewski space, part 2, Eran Stein
Eran Stein
-
Place: Building 502, Room 9

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.

- , The Ostaszewski space, part 1, Eran Stein
Eran Stein
-
Place: Building 502, Room 9

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.

- , The Erdos-Dushnik-Miller theorem revisited, part 2, Roy Shalev
Roy Shalev
-
Place: Building 502, Room 9

This is part II of last week's talk.

- , The Erdos-Dushnik-Miller theorem revisited, part 1, Roy Shalev
Roy Shalev
-
Place: Building 502, Room 9

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.

- , Graphs with no unfriendly partitions, Ron Langberg
Ron Langberg
-
Place: Building 502, Room 9

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.

- , A Dowker space from a ladder system, Shahak Shama
Shahak Shama
-
Place: Building 502, Room 9

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.

- , How to construct a Souslin tree the right way, part 2, Ari Brodsky (BIU)
Ari Brodsky (BIU)
-
Place: Building 502, Room 9

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

- , How to construct a Souslin tree the right way, part 1, Ari Brodsky (BIU)
Ari Brodsky (BIU)
-
Place: Building 502, Room 9

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

- , Infinite trees and partition relations, Dani Livne
Dani Livne
-
Place: building #502, room #9

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.

- , The Galvin-Hajnal formula and its applications to Cardinal Arithmetic, Luis Pereira (Lisbon)
Luis Pereira (Lisbon)
-
Place: Building 502, Room 9

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.

- , Coherent omission of intervals: Menger's and Hurewicz's problems, Piotr Szewczak (BIU and Cardinal Stefan Wyszyński University, Warsaw)
Piotr Szewczak (BIU and Cardinal Stefan Wyszyński University, Warsaw)
-
Place: Department room

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.
- , Putting a diamond inside the square, Assaf Rinot
Assaf Rinot
-
Place: seminar room

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

- , Martin's Axiom, and a strengthening of the Dushnik-Miller partition relation, Dani Livne
Dani Livne
-
Place: seminar room
- , Around a question of Bonanzinga and Matveev, Shir Sivroni
Shir Sivroni
-
Place: Seminar room

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.

- , Variants and generalizations of Diamond^*, Efrat Taub (BIU)
Efrat Taub (BIU)
-
Place: Room 201, Building 216

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.

 

- , The P-hierarchy of ultrafilters, Michal Machura (BIU)
Michal Machura (BIU)
-
Place:

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.

- , More on Resolvability of topological spaces, Nir Hakeyni (BIU)
Nir Hakeyni (BIU)
-
Place:

 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.

- , Resolvability of topological spaces, Nir Hakeyni (BIU)
Nir Hakeyni (BIU)
-
Place:

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)
-
Place:

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

 

Bibliograpy

 

, Avoiding rational distances, Ashutosh Kumar (HUJI)
Ashutosh Kumar (HUJI)
Place:

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.
 
, Well-colorings and the Hanf number for amalgamation, Chris Lambie-Hanson (HUJI)
Chris Lambie-Hanson (HUJI)
Place:

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
Place:

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

, Luzin sets and generalizations, Assaf Rinot
Assaf Rinot
Place:

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

, Nonuniversal colorings in ZFC, Dani Livne
Dani Livne
Place:

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.

, Large Sets, Assaf Rinot
Assaf Rinot
Place:

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

, Infinite-dimensional Jonsson algebras, Tomer Bauer
Tomer Bauer
Place:

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.

 

, Strong colorings without nontrivial polychromatic sets, Michal Machura
Michal Machura
Place:

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

, Infinite-dimensional polychromatic colorings, Dani Livne
Dani Livne
Place:

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

, Polychromatic colorings of the first uncountable cardinal, Dani Livne (BGU)
Dani Livne (BGU)
Place:

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

, From colorings to topology, Yuval Hachatrian
Yuval Hachatrian
Place:

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

, From topology to colorings, Yuval Hachatrian
Yuval Hachatrian
Place:

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

 

Lecture notes

, Anti-Ramsey colorings of the rational numbers, part 2, Lidor Eldabah
Lidor Eldabah
Place:

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

, Anti-Ramsey colorings of the rational numbers, part 1, לידור אלדבח
לידור אלדבח
Place:

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

, When Ramsey's theorem fails, אסף רינות
אסף רינות
Place:

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.