Set Theory

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We prove that the weakest form of Sierpinski’s onto mapping principle may consistently fail at the second uncountable cardinal.

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In the 1960s, Higgs showed that under GCH, any two bases of a matroid have the same size. More recently, Bowler and Geschke proved that this fails assuming Martin's Axiom (MA_{w1}, to be precise). We present these results without assuming any prior knowledge of matroid theory.

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We present a dichotomy statement concerning a subclass of transitive lists which is a consequence of Martin's Axiom and in fact follows both from K'_2 and from the Martin Axiom for Y-cc posets.

In a famous paper by Ben David and Magidor, they show the existence of an (\aleph_0,\aleph_\omega)-indecomposable ultrafilter on \aleph_{\omega+1}, in an intermediate model of a forcing that collapses \aleph_{\omega+1}. This is an indication that an (\aleph_0,\aleph_\omega)-indecomposable ultrafilter on \aleph_{\omega+1} might be enough to collapse \aleph_{\omega+1} by a forcing extension.
I will show that if \lambda is a singular cardinal, strong limit and there is an (\aleph_0,\lambda)-indecomposable ultrafilter on \lambda^+, then there is a \lambda^{++} c.c. forcing notion that collapses \lambda^+ and preserves \lambda.
As a corollary we will get that there can't be a (\aleph_0,\lambda)-indecomposable ultrafilter on  \lambda^+ if cf(\lambda)>\omega_1, in partcular there is no  (\aleph_0,\aleph_{\omega_2})-indecomposable ultrafilter on \aleph_{\omega_2+1}.

We revisit Shelah's 1976 construction of a Countryman line

We study various equivalent forms of weak square, proving in particular that the existence of a mu^+-Countryman line yields a special mu^+-Aronszajn tree.

In her master thesis, Yadai gave a proxy-based construction of a lambda^+-tree whose n^th power is Souslin and (n+1)^th power is Special. Here, we generalize this result to inaccessible cardinals. The heart of the matter is finding a consistent instance of the proxy that enables this generalization. We shall formulate this instance and prove its consistency.

This is joint work with Brodsky and Yadai.

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We shall present results from Part III of our series of papers with Tanmay Inamdar, entitled Was Ulam right?.
This part centers around indecomposable ideals as already announced in a talk from a year ago (video).
As an application, we shall improve Conclusion 4.8(2) of [Sh:365], increasing the number of colors to be as large as possible (theta=lambda in Shelah's notation).

We prove that for every regular kappa above Aleph_1, if square(kappa) holds, then there exists a C-sequence over kappa for which the corresponding T(rho_0) satisfies a strong negative partition relation (see Definition 2.9 of our preprint). To mitigate for the lack of a nonreflecting stationary set, the proof goes through a division into four cases.

We show how to deal with the case where the vanishing levels of a C-sequence do not cover a club. As an application, we prove that the conclusion of Theorem D from our preprint holds true also for kappa that is a subtle cardinal. Time permits, we shall also point out how to get a long decreasing and increasing chains of Aronszajn lines.

We prove that the conclusion of Theorem A from our preprint holds true also for mu an ineffable cardinal. In particular, for every regular uncountable cardinal mu, if there is a special Aronszajn tree of height mu^+, then there is one satisfying a very strong negative partition relation.

We prove that for every strongly inaccessible cardinal lambda, any basis for the class of special lambda^+-Aronszajn lines must have size at least lambda.

We prove that for every regular uncountable cardinal lambda, if there exists a lambda^+-Aronszajn line, then there exists one with no lambda^+-Countryman subline. We have already seen the proof for the case that is lambda is non-ineffable. Today, we shall give a separate proof for ineffables.

Let lambda be a regular uncountable cardinal that is non-ineffable, and consider the class C of all special lambda^+-Aronszajn trees ordered by the relation of weak embeddability. We shall prove that the if above class is nonempty then it moreover contains a copy of (lambda^+,\subseteq).

We complete the proof that if there exists an w2-Aronszajn line, then there exists one that contains no w2-Countryman line. This provides a negative answer to a question of Justin Moore

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Aronszajn (1935) proved that there exists a special w1-Aronszajn tree and Jensen (1972) proved that, more generally, the existence of a special (mu^+)-Aronszajn tree is equivalent to the weak square principle holding at mu. We shall present a far-reaching extension of Jensen's theorem. As an application, we get that if there exists a special w2-Aronszajn tree, then any basis for the class of w2-Aronszajn lines must be of the maximal possible size 2^{w2}. This is in contrast with Justin Moore's theorem (2006) that the Proper Forcing Axiom implies that the class of w1-Aronszajn lines admits a basis of size 2.

The proof combines walks on ordinals, club guessing, strong colorings, vanishing levels of trees and a bit of finite combinatorics. The results are not limited to w2 (or to successor cardinals).

This is joint work with Tanmay Inamdar.

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We discuss generalizations to w2 of Moore's principle Mho and prove the consistency of the failure of one of them. This answers a question of Inamdar and Rinot.

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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.

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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.
 

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

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

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

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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

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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.

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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.

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

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.

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

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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.

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.

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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.

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.

We'll discuss our recent preprint.

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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.

The last talk for this series

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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.

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 .

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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

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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).

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.

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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.
 

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.

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.

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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.

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.

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.

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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.

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.

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.

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)

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

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.

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.

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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.

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.

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.

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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.

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.

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

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

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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.
 

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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.

last talk for this series

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We discuss colorings of triples

Results on the Generic Kurepa Hypothesis

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We discuss consistency results concerning Kurepa trees

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.

Link to recording.

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.

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

Link to video recording.

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.

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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.

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.

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

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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 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.

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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^++).

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).

Joint work with Itay Kaplan and Saharon Shelah.

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 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. 

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.

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We'll talk about colorings of triples of w2

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.

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

Recoding is now available.

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

Covering Neeman's proof

Attached

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. 

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

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 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.

(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.

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.

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.

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.

Let p:[\kappa]^2\to \theta be a partition of all unordered pairs from a cardinal \kappa to \theta pieces. A coloring f:[\kappa]^2\to \lambda is *strong over *p if for every A\subseteq \kappa with |A|=\kappa there is some i=i(A)<\theta such that ran (f\restriction ([A]^2\cap p^{-1}(\{i\}))=\lambda.

The partition symbol for asserting the existence of a strong \lambda-coloring on \kappa over a partition p is
\kappa\not\longrightarrow_p[\kappa]^2_\lambda.

In the talk we shall define more strong coloring symbols, like Pr_1 and Pr_0 and sketch the proofs of the following results:

1. Strong colorings over finite partitions exist in ZFC whenever they exist without partitions.
2. Instances of the GCH and of the SCH imply the existence of strong colorings over infinite partitions.
3. Whether for every countable partition of p:[\omega_1]^2\to \omega there is a strong \aleph_1-coloring over it, is independent over ZFC + \neg CH.

These are joint results with Bill Chen and Juris Steprans.

Recoding is now available.

I will discuss the paper of 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.

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. 

I will discuss the paper of Shelah from the title and results which can be proved using similar techniques. A sample result is that one cannot have a chain of subsets of Omega2 strongly increasing modulo finite. The negative results we discuss can be contrasted with positive results of Koszmider on strong chains and results of Baumgartner on strongly almost disjoint families. The paper of Shelah is very short and relies only on a basic understanding of set theory (regular cardinals, closed unbounded sets). 

In the talk we will focus on compactness principles at the double successor of a regular cardinal kappa. We start by showing that if kappa^{<kappa} =kappa and lambda>kappa is a weakly compact cardinal, then in the Mitchell model V[M(kappa,lambda)] the tree property at kappa^{++} is indestructible under all kappa^+-cc forcing notions which live in the intermediate submodel V[Add(kappa,lambda)]. This result has direct applications for Prikry-style forcing notions and hence for the tree property at the double successor of a singular strong limit cardinal (it simplifies existing results and can be used to prove new results). Then we will discuss stationary reflection and its variants and the indestructibility under kappa^+-cc forcing notions.

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.

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.

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.

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.

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.

Abstract is attached

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.

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.

This is joint work with Miguel Moreno and Assaf Rinot.

This is a continuation of the lecture from last week.

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.

this is a continuation of our previous talk

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.

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.

We formulate a weakening of Ostaszewski's club principle, and show that it holds assuming the existence of a Souslin tree. In the next talk, we shall show that our principle suffices for the construction of an Ostaszewski space.

Local club condensation (LCC) is an abstract formulation of a condensation property that canonical L-like models have. It was introduced by Friedman and Holy. Diamond^{sharp} is a strengthening of Jensen's diamond principle which was introduced by Devlin.
We shall show that diamond^{sharp} follows from LCC and the existence of a suitable Delta_1 definable well order, and then present an application of diamond^{sharp} to Generalized Descriptive Set Theory (GDST).

This is joint work with Miguel Moreno and Assaf Rinot

The Minimal Tower Problem was one of most famous question in Cardinal Invariants. We will present a combinatorial argument of this proof, which without using model theory and forcing, motivated by Malliaris and Shelah's proof.

In the 1970'ies, Bukowský identified a beautiful and handy criterion for when V is a forcing extension of a given inner model, which proved very useful recently in set theoretical geology. In the 1990'ies, Woodin isolated his extender algebra which makes use of a large cardinal, a Woodin cardinal. It turns out that Bukowský's theorem and Woodin's extender algebra may be presented in a uniform fashion - one proof and one forcing gives both results. We will present the proof and then discuss its application in inner model theoretic geology. This is joint work with Grigor Sargsyan and Farmer Schlutzenberg.

During this talk we will discuss where in the generalized Borel-reducibility hierarchy are the isomorphism relation of first order complete theories. These theories are divided in two kind:classifiable and non-classifiable. To study the classifiable theories case is needed the use of Ehrenfeucht-Fraïssé games. On the other hand the study of the non-classifiable theories is done by using colored trees. The goal of the talk is to see the classifiable theories case and start the non-classifiable theories case by proving that it is possible to map every element of the generalized Baire, f, into a colored tree, J(f), such that; for every f and g elements of the generalized Baire space, J(f) and J(g) are isomorphic as colored trees if and only if f and g coincide on a club.

An infinite cover of a topological space is an w-cover if every finite subset of this space is contained in some member of the cover, and the whole space is not a member the cover. A cover of a topological space is a gamma-cover if every point of this space belongs to all but finitely many members of this cover. A gamma-space is a space in which every open w-cover contains a gamma-cover.
In this talk, we will present the details how to construct it.

 We will study infinite two player games and the large
 cardinal strength corresponding to their determinacy. For games of
 length $\omega$ this is well understood and there is a tight
 connection between the determinacy of projective games and the
 existence of canonical inner models with Woodin cardinals. For games
 of arbitrary countable length, Itay Neeman proved the determinacy of
 analytic games of length $\omega \cdot \theta$ for countable $\theta
 \> \omega$ from a sharp for $\theta$ Woodin cardinals.

We aim for a converse at successor ordinals. In joint work with Juan
 P. Aguilera we showed that determinacy of $\boldsymbol\Pi^1\_{n+1}$
 games of length $\omega^2$ implies the existence of a premouse with
 $\omega+n$ Woodin cardinals. This generalizes to a premouse with
 $\omega+\omega$ Woodin cardinals from the determinacy of games of length
 $\omega^2$ with $\Game^{\mathbb{R}}\boldsymbol\Pi^1\_1$ payoff.

If time allows, we will also sketch how these methods can be adapted
to, in combination with results of Nam Trang, obtain $\omega^\alpha+n$ Woodin
cardinals for countable ordinals $\alpha$ and natural numbers $n$ from
the determinacy of sufficiently long projective games.

An uncountable cardinal κ is Jonnson if only if the set of proper subsets of κ that are of cardinality κ is stationary. Though this property has large cardinal strength it is not at all clear that Jonnson cardinals do in fact need to be large in the obvious sense. For example, it is known that Jonsson cardinals can be singular. 

In this talk we will use the methods of Inner Model Theory to show that, given the assumption that the least singular cardinal is Jonsson, there is a canonical model with a strong cardinal together with a class of Silver indiscernibles for this model. (The proof presented will make some simplifying assumptions.) Time permitting, we may discuss approaches to extend this result to show the existence of inner models with Woodin cardinals and more.

Sierpinski's now classical result states that there is an edge 2-colouring of the complete graph on aleph1 vertices so that there are no uncountable monochromatic subgraphs. In the 1970s, Erdos, Galvin and Hajnal asked what other graphs with large chromatic number admit similar edge colourings i.e., with no 'large' monochromatic subgraphs. We plan to review some recent advances on this problem and in particular, connect the question to Shelah's ladder system uniformization theory.

Last week, we gave a detailed proof of Lemma 1.13 from the notes:
http://u.math.biu.ac.il/~morenom3/GDST-2018.pdf
This week, we shall continue, proving that, if V=L, then $\kappa$-Borel* class is equal to the $\Sigma1^ 1(\kappa)$ class.

After introducing the notions of $\kappa$-Borel class, $\kappa$-$\Delta_1^1$ class, $\kappa$-Borel^* class in the previous talk ( http://u.math.biu.ac.il/~morenom3/GDST-2018.pdf ), in this talk, we will show the relation between this classes.
In descriptive set theory the Borel class, the $\Delta_1^1$ class, the Borel* class are the same class, this doesn't hold in the generalized descriptive set theory, in particular under the assumption V=L the Borel* class is equal to the $\Sigma1^ 1$ class.

How big can countable unions of countable sets be? Assuming the axiom of choice, countable. Not assuming the axiom of choice, it is not hard to arrange situation where there are many incomparable cardinals which are the countable union of countable sets. But none of them are "particularly large". While a countable union of countable sets can at most be mapped onto \omega_1, its power set can be made much larger. We prove an old (and nearly forgotten) theorem of Douglass Morris, that it is consistent that for every \alpha there is a set which is a countable union of countable sets, but its power set can be mapped onto \alpha.

This is the first of many of talks in which an overview of the Borel-reducibility hierarchy in the generalized Baire space will be given. The aim of this talk is to introduce the notions of $\kappa$-Borel class, $\kappa$-$\Delta_1^{^1}$ class, $\kappa$-Borel^* class, and show the relation between these classes.

We discuss a joint work with Unger about stationary reflection at subcompact cardinals and the consistency of stationary reflection at successor of singular cardinal from a $Pi^1_1$-$kappa^+$-subcompact cardinal.

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.

The consistency of the Chang's Conjecture (CC) variant (aleph_{omega + 1}, aleph_omega) ->> (aleph_2, aleph_1) is a major open question. A combinatorial consequence of this instance of CC is the existence of a certain strongly increasing sequence, of length aleph_2, of functions from omega to some fixed ordinal below omega_2 (we are calling such a sequence a club-increasing sequence). In response to a question from the speaker, Paul Larson proved the consistency of the existence of a club-increasing sequence of length aleph_2 using a P_max forcing variation. In this talk, we will prove some basic results about club-increasing sequences, including the facts that their existence follows from the relevant instance of CC and that club-increasing sequences of length omega_n do not exist for n at least 4. We will then give an introduction to P_max forcing, in preparation for a future talk in which we will present a sketch of Larson's proof.

We work with a $\lambda$-frame, which is an abstract elementary class endowed with a collection of basic types and a non-forking relation satisfying certain natural properties with respect to models of cardinality $\lambda$.

We will show that assuming the diamond axiom, any basic type admits a non-forking extension that has a uniqueness triple.
Prior results of Shelah in this direction required either some form of diamond at two consecutive cardinals, or a constraint on the number of models of size $\lambda$.

This is joint work of with Adi Jarden.

Strongly compact cardinals are characterized by the property that  any $\kappa$-complete filter can be extended to a $\kappa$-complete ultrafilter. When restricting the cardinality of the underlying set, we obtain a nontrivial hierarchy. For example, when requiring the extension property to hold only for filters on $\kappa$, we obtain Gitik's $\kappa$-compact cardinals, which are known to be consistently weaker than $\kappa$ being $\kappa^+$-strongly compact.

In this talk I will focus on the level by level connection between the filter extension property and the compactness for $L_{\kappa,\kappa}$. Using the compactness, I will show that if $\kappa$ is $\kappa$-compact then $\square(\kappa^{+})$-fails.

In the 1970's, consistent examples of k-cc posets whose square is not k-cc were constructed by Laver, Fleissner, and Galvin. Later on, ZFC examples were constructed by Todorcevic, Shelah and others. The hardest case, being k=w2, was resolved by Shelah in 1997.
In this work, we obtain analogous results for k-Knaster posets. Among others, for any successor cardinal k, we produce a ZFC example of a k-Knaster poset whose w-power is not k-cc.
To do so, we introduce a new coloring principle, and establish the existence of various instances of it.
We also introduce a new cardinal invariant for k, denoted chi(k), that, roughly speaking, measures how far k is from being weakly compact. It is proved that by forcing over a model with a weakly compact cardinal k, chi(k) could be made equal to any prescribed regular cardinal <= k.
Further byproducts of this work show that the main results of [1] and [2] are sharp.

This is joint work with Chris Lambie-Hanson.

[1] A. Rinot, Transforming rectangles into squares, with applications to strong colorings, Adv. Math., 231(2): 1085-1099, 2012.
[2] A. Rinot, Complicated colorings, Math. Res. Lett., 21(6): 1367–1388, 2014.

We'll continue the discussion on some asymptotic variants of the club principle - the limsup versions. Joint work with Shelah. Here's the preperint: http://www.math.huji.ac.il/~akumar/svcp.pdf

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

Continuing work from the previous talk, we sketch the proofs of some definable versions of Hall's matching theorem.  Then we apply them to get various geometric paradoxes with definable pieces including our recent Borel circle squaring result with Andrew Marks.

Last week, we presented an approach for constructing a Dowker space of size contniuum, ending up with a statement of a lemma that would yield such a space. In this talk, we shall prove this lemma.

Lecture notes may be found in here.

A topological space is said to be Dowker if it is normal but its product with the unit interval is not normal. In this lecture, we shall present a construction, due to Balogh, of a Dowker space of size continuum.

Lecture notes may be found in here.

We construct a model of ZF with an uncountable set of reals having a unique condensation point. This answers a question of Sierpinski from 1918.

In this talk I will define the notion of Magidor Cardinal (\omega bounded Jonsson cardinal) which is a generalization of Jonsson cardinal. I will show that the analog of Jonsson filter for Magidor cardinals is inconsistent with ZFC. This lecture is based on a joint work with Shimon Garti and Saharon Shelah

We shall present a new method for obtaining the tree property at w2 from the consistency of a weakly compact cardinal.
The method is due to Stejskalova, and uses Grigorieff forcing.
 

the abstract may be found in here.

We address the question whether, assuming the generalized continuum hypothesis (GCH), the existence of a k-Aronszajn tree entails the existence of a distributive one.

Biblography: see here.

Fremlin asked if the null ideal restricted to a non null set of reals could be isomorphic to the non stationary ideal on omega_1. Eskew asked if the null and the meager ideal could both be somewhere countably saturated. We'll show that the answer to both questions is yes. Joint work with Shelah.

This is part 2 of last week's talk.

We are going to continue the analysis of generalised scattered orders, proving the result described towards the end of Chris Lambie-Hanson’s talk. This states that consistently, for every sigma-scattered linear ordering there is a colouring of its pairs in black & white such that every triple contains a white pair and every copy of one of the following order-types contains a black pair:

• omega_1^omega
• (omega_1^omega)^*
• omega_1 * (omega * omega^*)^omega
• omega_1^* * (omega * omega^*)^omega
• (omega * omega^*)^omega * omega_1
• (omega * omega^*)^omega * omega_1^*

This generalises a 46-year-old Theorem of Erdős & Rado about ordinals. A sufficient hypothesis implying this theorem is the existence of a colouring of the pairs of omega_1 * omega in black & white such that every triple contains a black pair and every subset of full order-type contains a white one. Time permitting we may present a proof that stick = b = Aleph_1 implies the existence of such a colouring. Here b is the unbounding number and stick = Aleph_1 is a weakening of the club principle which was considered by Baumgartner 41 years ago, named by Broverman, Ginsburg, Kunen & Tall two years thereafter and twenty years ago reconsidered as a cardinal characteristic by Fuchino, Shelah & Soukup.

The class of scattered linear orders, isolated by Hausdorff, plays a prominent role in the study of general linear orders. In 2006, Dzamonja and Thompson introduced classes of orders generalizing the class of scattered orders. For a regular cardinal kappa, they defined the classes of kappa-scattered and weakly kappa-scattered linear orders. For kappa = omega, these two classes coincide and are equal to the classical class of scattered orders. For larger values of kappa, though, the two classes are provably different. In this talk, we will investigate properties of these generalized scattered orders with respect to partition relations, in particular the extent to which the classes of kappa-scattered or weakly kappa-scattered linear orders of size kappa are closed under partition relations of the form tau -> (phi, n) for n < omega. We will show that, assuming kappa^{<kappa} = kappa, the class of weakly kappa-scattered orders is closed under all such partition relations while, for uncountable values of kappa, the class of kappa-scattered orders consistently fails to be closed. Along the way, we will prove a generalization of the Milner-Rado paradox and look at some results regarding ordinal partition relations. This is joint work with Thilo Weinert.

This is an expository presentation following the paper "Minimality of non $\sigma$-scattered orders" by Ishiu and Moore. In the first part of the talk we will introduce the invariant $\Omega(L)$ of a linear order $L$, and characterize $\sigma$-scattered linear orders in terms of this invariant. In the second part, we will prove under the forcing axiom $\mathsf{PFA}^+$ that any linear order which is minimal with respect to embedding among the non $\sigma$-scattered orders must be either a real or Aronszajn type.

This is an expository presentation following the paper "Minimality of non $\sigma$-scattered orders" by Ishiu and Moore. In the first part of the talk we will introduce the invariant $\Omega(L)$ of a linear order $L$, and characterize $\sigma$-scattered linear orders in terms of this invariant. In the second part, we will prove under the forcing axiom $\mathsf{PFA}^+$ that any linear order which is minimal with respect to embedding among the non $\sigma$-scattered orders must be either a real or Aronszajn type.

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

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

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

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.

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

The paper may be found in here.

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

The paper may be found in here.

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

The results are taken from the following paper.

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

The results are taken from the following paper.

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

The results are taken from the following paper.

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

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

The lecture will be based on the following paper.

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

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

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

This is part II of last week's talk.

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

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

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

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

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

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

Lecture notes.

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

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

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

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

Bibliography

Bibliography

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

Following Bonanzinga and Matveev, this question is considered here from a combinatorical point of view.  We give an answer to a problem thay have stated,  and present some related open problems.
 
All is taken from this paper by Boaz Tsaban.
The slides are available here.

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

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

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

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