Complete list of blog posts

  • Square with built-in diamond-plus Joint work with Ralf Schindler. Abstract. We formulate combinatorial principles that combine the square principle with various strong forms of diamond, and prove that the strongest amongst them holds in $L$ for every infinite cardinal. As an application, we prove that the following two hold in $L$: For every infinite regular cardinal $\lambda$, there exists a special $\lambda^+$-Aronszajn tree ...
  • The vanishing levels of a tree Joint work with Shira Yadai and Zhixing You. Abstract. We initiate the study of the spectrum $Vspec(\kappa)$ of sets that can be realized as the vanishing levels $V(\mathbf T)$ of a normal $\kappa$-tree $\mathbf T$. The latter is an invariant in the sense that if $\mathbf T$ and $\mathbf T’$ are club-isomorphic, then $V(\mathbf T)\bigtriangleup V(\mathbf T’)$ ...
  • Proxy principles in combinatorial set theory Joint work with Ari Meir Brodsky and Shira Yadai. Abstract. The parameterized proxy principles were introduced by Brodsky and Rinot in a 2017 paper as new foundations for the construction of $\kappa$-Souslin trees in a uniform way that does not depend on the nature of the (regular uncountable) cardinal $\kappa$. Since their introduction, these principles have facilitated ...
  • Reduced powers of Souslin trees Joint work with Ari Meir Brodsky. Abstract. We study the relationship between a $\kappa$-Souslin tree $T$ and its reduced powers $T^\theta/\mathcal U$. Previous works addressed this problem from the viewpoint of a single power $\theta$, whereas here, tools are developed for controlling different powers simultaneously. As a sample corollary, we obtain the consistency of an $\aleph_6$-Souslin tree $T$ ...
  • Inclusion modulo nonstationary Joint work with Gabriel Fernandes and Miguel Moreno. Abstract. A classical theorem of Hechler asserts that the structure $\left(\omega^\omega,\le^*\right)$ is universal in the sense that for any $\sigma$-directed poset $\mathbb P$ with no maximal element, there is a ccc forcing extension in which $\left(\omega^\omega,\le^*\right)$ contains a cofinal order-isomorphic copy of $\mathbb P$. In this paper, we prove ...
  • Fake Reflection Joint work with Gabriel Fernandes and Miguel Moreno. Abstract. We introduce a generalization of stationary set reflection which we call filter reflection, and show it is compatible with the axiom of constructibility as well as with strong forcing axioms. We prove the independence of filter reflection from ZFC, and present applications of filter reflection to the ...
  • Perspectives on Set Theory, November 2023 I gave an invited talk at the Perspectives on Set Theory conference, November 2023. Talk Title: May the successor of a singular cardinal be Jónsson? Abstract: We’ll survey what’s known about the question in the title and collect ten open problems. Downloads:
  • Diamond on ladder systems and countably metacompact topological spaces Joint work with Rodrigo Rey Carvalho and Tanmay Inamdar. Abstract. Leiderman and Szeptycki proved that a single Cohen real introduces a ladder system $L$ over $\aleph_1$ for which the 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 ...
  • Ordinal definable subsets of singular cardinals Joint work with James Cummings, Sy-David Friedman, Menachem Magidor, and Dima Sinapova. Abstract. A remarkable result by Shelah states that if $\kappa$ is a singular strong limit cardinal of uncountable cofinality then there is a subset $x$ of $\kappa$ such that $\text{HOD}_x$ contains the power set of $\kappa$. In this paper, we develop a version of diagonal ...
  • Diamond on Kurepa trees Joint work with Ziemek Kostana and Saharon Shelah. Abstract. We introduce a new weak variation of diamond that is meant to only guess the branches of a Kurepa tree. We demonstrate that this variation is considerably weaker than diamond by proving it is compatible with Martin’s axiom. We then prove that this principle is nontrivial by ...
  • 120 Years of Choice, July 2024 I gave an invited talk at the 120 Years of Choice conference, July 2024. Talk Title: Mathematician’s best friend Abstract: Jensen’s diamond is a very useful postulate. It is well-known that it implies the continuum hypothesis, that it is equivalent to seemingly weaker forms in which the frequency of guesses is not imposed to be stationary or ...
  • Full Souslin trees at small cardinals Joint work with Shira Yadai and Zhixing You. Abstract. A $\kappa$-tree is full if each of its limit levels omits no more than one potential branch. Kunen asked whether a full $\kappa$-Souslin tree may consistently exist. Shelah gave an affirmative answer of height a strong limit Mahlo cardinal. Here, it is shown that these trees may ...
  • A club guessing toolbox I Joint work with Tanmay Inamdar. Abstract. Club guessing principles were introduced by Shelah as a weakening of Jensen’s diamond. Most spectacularly, they were used to prove Shelah’s ZFC bound on the power of the first singular cardinal. These principles have found many other applications: in cardinal arithmetic and PCF theory; in the construction of combinatorial objects on ...
  • The failure of diamond on a reflecting stationary set Joint work with Moti Gitik. Abstract: It is shown that the failure of $\diamondsuit_S$, for a subset $S\subseteq\aleph_{\omega+1}$ that reflects stationarily often, is consistent with GCH and $\text{AP}_{\aleph_\omega}$, relatively to the existence of a supercompact cardinal. This should be comapred with a theorem of Shelah, that GCH and $\square^*_\lambda$ entails $\diamondsuit_S$ for any subset $S\subseteq\lambda^+$ that reflects stationarily often. We ...
  • A relative of the approachability ideal, diamond and non-saturation Abstract: Let $\lambda$ denote a singular cardinal. Zeman, improving a previous result of Shelah, proved that $\square^*_\lambda$ together with $2^\lambda=\lambda^+$ implies $\diamondsuit_S$ for every $S\subseteq\lambda^+$ that reflects stationarily often. In this paper, for a subset $S\subset\lambda^+$, a normal subideal of the weak approachability ideal is introduced, and denoted by $I$. We say that the ideal is fat if it ...
  • Review: Stevo Todorcevic’s CRM-Fields-PIMS Prize Lecture After winning the 2012 CRM-Fields-PIMS Prize, Stevo Todorcevic gave a series of talks on his research: at CRM, at PIMS and at the Fields Institute. The director of the Fields Institute asked me to write a short review on Stevo’s Fields lecture for the Winter 2013 Fields Notes, and I thought I should post my review ...
  • May the successor of a singular cardinal be Jonsson? Abstract: We collect necessary conditions for the successor of a singular cardinal to be Jónsson. Downloads:
  • Logic in Hungary, August 2005 These are the slides of a contributed talk given at the Logic in Hungary 2005 meeting (Budapest, 5–11 August 2005). Talk Title: On the consistency strength of the Milner-Sauer Conjecture Abstract: In their paper from 1981, after learning about Pouzet‘s theorem that any poset of singular cofinality mush contain an infnite antichain, Milner and Sauer came up with the following ...
  • Annual conference of the IMU, May 2006 This talk was given at the 2006 Annual Conference of the Israel Mathematical Union (Neve Ilan, 25-26 May 2006). Talk Title:  The Milner-Sauer conjecture and covering numbers Abstract: In their paper from 1981, after learning about Pouzet‘s theorem that any poset of singular cofinality mush contain an infnite antichain, Milner and Sauer came up with the following conjecture: Every poset $\mathbb P$ of singular ...
  • The chromatic numbers of the Erdos-Hajnal graphs Recall that a coloring $c:G\rightarrow\kappa$ of an (undirected) graph $(G,E)$ is said to be chromatic if $c(v_1)\neq c(v_2)$ whenever $\{v_1,v_2\}\in E$. Then, the chromatic number of a graph $(G,E)$ is the least cardinal $\kappa$ for which there exists a chromatic coloring $c:G\rightarrow\kappa$. Motivated by a paper I found yesterday in the arXiv, this post will be ...
  • Partitioning a reflecting stationary set Joint work with Maxwell Levine. Abstract. We address the question of whether a reflecting stationary set may be partitioned into two or more reflecting stationary subsets, providing various affirmative answers in ZFC. As an application to singular cardinals combinatorics, we infer that it is never the case that there exists a singular cardinal all of whose scales ...
  • A microscopic approach to Souslin-tree constructions. Part II Joint work with Ari Meir Brodsky. Abstract. In Part I of this series, we presented the microscopic approach to Souslin-tree constructions, and argued that all known $\diamondsuit$-based constructions of Souslin trees with various additional properties may be rendered as applications of our approach. In this paper, we show that constructions following the same approach may be carried ...
  • A guessing principle from a Souslin tree, with applications to topology Joint work with Roy Shalev. Abstract. We introduce a new combinatorial principle which we call $\clubsuit_{AD}$. 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 $\clubsuit_{AD}$ follow ...
  • Weak square and stationary reflection Joint work with Gunter Fuchs. Abstract. It is well-known that the square principle $\square_\lambda$ entails the existence of a non-reflecting stationary subset of $\lambda^+$, whereas the weak square principle $\square^*_\lambda$ does not. Here we show that if $\mu^{cf(\lambda)}<\lambda$ for all $\mu<\lambda$, then $\square^*_\lambda$ entails the existence of a non-reflecting stationary subset of $E^{\lambda^+}_{cf(\lambda)}$ in the forcing extension for ...
  • Souslin trees at successors of regular cardinals Abstract. We present a weak sufficient condition for the existence of Souslin trees at successor of regular cardinals. The result is optimal and simultaneously improves an old theorem of Gregory and a more recent theorem of the author. Downloads: Citation information: A. Rinot, Souslin trees at successors of regular cardinals, Math. Log. Q., 65(2): 200-204, 2019.
  • Higher Souslin trees and the GCH, revisited Abstract.  It is proved that for every uncountable cardinal $\lambda$, GCH+$\square(\lambda^+)$ entails the existence of a $\text{cf}(\lambda)$-complete $\lambda^+$-Souslin tree. In particular, if GCH holds and there are no $\aleph_2$-Souslin trees, then $\aleph_2$ is weakly compact in Godel’s constructible universe, improving Gregory’s 1976 lower bound. Likewise, if GCH holds and there are no $\aleph_2$ and $\aleph_3$ Souslin trees, ...
  • More notions of forcing add a Souslin tree Joint work with Ari Meir Brodsky. Abstract.   An $\aleph_1$-Souslin tree is a complicated combinatorial object whose existence cannot be decided on the grounds of ZFC alone. But 15 years after Tennenbaum and independently Jech devised notions of forcing for introducing such a tree, Shelah proved that already the simplest forcing notion — Cohen forcing — adds an ...
  • Putting a diamond inside the square Abstract. By a 35-year-old theorem of Shelah, $\square_\lambda+\diamondsuit(\lambda^+)$ does not imply square-with-built-in-diamond_lambda for regular uncountable cardinals $\lambda$. Here, it is proved that $\square_\lambda+\diamondsuit(\lambda^+)$ is equivalent to square-with-built-in-diamond_lambda for every singular cardinal $\lambda$. Downloads: Citation information: A. Rinot, Putting a diamond inside the square, Bull. Lond. Math. Soc., 47(3): 436-442, 2015.
  • Same Graph, Different Universe Abstract. May the same graph admit two different chromatic numbers in two different universes? how about infinitely many different values? and can this be achieved without changing the cardinals structure? In this paper, it is proved that in Godel’s constructible universe, for every uncountable cardinal $\lambda$ below the first fixed-point of the $\aleph$-function, there exists a ...
  • On the ideal J[kappa] Abstract. Motivated by a question from a recent paper by Gilton, Levine and Stejskalova, we obtain a new characterization of the ideal $J$, from which we confirm that $\kappa$-Souslin trees exist in various models of interest. As a corollary we get that for every integer $n$ such that $\mathfrak b<2^{\aleph_n}=\aleph_{n+1}$, if $\square(\aleph_{n+1})$ holds, then there exists ...
  • A new small Dowker space Joint work with Roy Shalev and Stevo Todorcevic. Abstract. It is proved that if there exists a Luzin set, or if either the stick principle or $\diamondsuit(\mathfrak b)$ hold, then an instance of the guessing principle $\clubsuit_{AD}$ holds at the first uncountable cardinal. In particular, any of the above hypotheses entails the existence of a Dowker space ...
  • A microscopic approach to Souslin-tree constructions. Part I Joint work with Ari Meir Brodsky. Abstract.  We propose a parameterized proxy principle from which $\kappa$-Souslin trees with various additional features can be constructed, regardless of the identity of $\kappa$. We then introduce the microscopic approach, which is a simple method for deriving trees from instances of the proxy principle. As a demonstration, we give a construction ...
  • A forcing axiom deciding the generalized Souslin Hypothesis Joint work with Chris Lambie-Hanson. Abstract. We derive a forcing axiom from the conjunction of square and diamond, and present a few applications, primary among them being the existence of super-Souslin trees. It follows that for every uncountable cardinal $\lambda$, if $\lambda^{++}$ is not a Mahlo cardinal in Godel’s constructible universe, then $2^\lambda = \lambda^+$ entails the ...
  • A remark on Schimmerling’s question Joint work with Ari Meir Brodsky. Abstract. Schimmerling asked whether $\square^*_\lambda$ together with GCH entails the existence of a $\lambda^+$-Souslin tree, for a singular cardinal $\lambda$. Here, we provide an affirmative answer under the additional assumption that there exists a non-reflecting stationary subset of $E^{\lambda^+}_{\neq cf(\lambda)}$. As a bonus, the outcome $\lambda^+$-Souslin tree is moreover free. Downloads: Citation information: A. ...
  • A Shelah group in ZFC Joint work with Márk Poór. Abstract. In a paper from 1980, Shelah constructed an uncountable group all of whose proper subgroups are countable. Assuming the continuum hypothesis, he constructed an uncountable group $G$ that moreover admits an integer $n$ satisfying that for every uncountable $X\subseteq G$, every element of $G$ may be written as a group ...
  • Distributive Aronszajn trees Joint work with Ari Meir Brodsky. Abstract.  Ben-David and Shelah proved that if $\lambda$ is a singular strong-limit cardinal and $2^\lambda=\lambda^+$, then $\square^*_\lambda$ entails the existence of a $\lambda$-distributive $\lambda^+$-Aronszajn tree. Here, it is proved that the same conclusion remains valid after replacing the hypothesis $\square^*_\lambda$  by $\square(\lambda^+,{<\lambda})$. As $\square(\lambda^+,{<\lambda})$ does not impose a bound on the ...
  • Mathematics Colloquium, Bar-Ilan University, November 2013 I gave a colloquium talk at Bar-Ilan University on November 10, 2013. Title: Forcing as a tool to prove theorems Abstract: Paul Cohen celebrated solution to Hilbert’s first problem showed that the Continuum Hypothesis is independent of the usual axioms of set theory. His solution involved a new apparatus for constructing models of set theory – the method of ...
  • Square principles Since the birth of Jensen’s original Square principle, many variations of the principle were introduced and intensively studied. Asaf Karagila suggested me today to put some order into all of these principles. Here is a trial. Definition. A square principle for a cardinal $\theta$ asserts the existence of a sequence $\Gamma=\langle C_\alpha \mid \alpha<\theta\rangle$ such that ...
  • Gdańsk Logic Conference, May 2023 I gave an invited talk at the first Gdańsk Logic Conference, May 2023. Talk Title: Was Ulam right? Abstract: An Ulam matrix is one of the earliest gems of infinite combinatorics. Around the same time of its discovery, another Polish mathematician, Wacław Sierpiński was toying with pathological consequences of the continuum hypothesis. We shall argue that one ...
  • A series of lectures on Club_AD, February–March 2023 As part of the Thematic Program on Set Theoretic Methods in Algebra, Dynamics and Geometry (Fields Institute, January–June, 2023), Spencer Unger and I delivered a Graduate Course on Set Theory, Algebra and Analysis. My part of the course was a series of lectures around the guessing principle $\clubsuit_{AD}$. The first lecture motivated the Dowker space problem ...
  • Winter School in Abstract Analysis, January 2023 I gave a 3-lecture tutorial at the Winter School in Abstract Analysis in Steken, January 2023. Title: Club guessing Abstract. Club guessing principles were introduced by Shelah as a weakening of Jensen’s diamond. Most spectacularly, they were used to prove Shelah’s ZFC bound on the power of the first singular cardinal. These principles have found many ...
  • Shelah’s approachability ideal (part 2) In a previous post, we defined Shelah’s approachability ideal $I^{<\lambda}$ such that for club many $\delta\in S$, the union $\bigcup_{\alpha<\delta}\mathcal D_\alpha$ contains all the initial segments of some small cofinal subset, $A_\delta$, of $\delta$. The ...
  • MFO workshop in Set Theory, January 2022 I gave an invited talk at the Set Theory meeting in Obwerwolfach, January 2022. Talk Title: A dual of Juhasz’ question Abstract: Juhasz asked whether $\clubsuit$ implies the existence of a Souslin tree. Here we settle the dual problem of whether a Souslin tree implies (a useful weakening of) $\clubsuit.$ This is joint work with Roy Shalev. Downloads:
  • Jensen’s diamond principle and its relatives This is chapter 6 in the book Set Theory and Its Applications (ISBN: 0821848127). Abstract: We survey some recent results on the validity of Jensen’s diamond principle at successor cardinals. We also discuss weakening of this principle such as club guessing, and anti-diamond principles such as uniformization. A collection of open problems is included. Table of Contents: Diamond Weak Diamond ...
  • Jones’ theorem on the cardinal invariant $\mathfrak p$ This post continues the study of the cardinal invariant $\mathfrak p$. We refer the reader to a previous post for all the needed background. For ordinals $\alpha,\alpha_0,\alpha_1,\beta,\beta_0,\beta_1$, the polarized partition relation $$\left(\begin{array}{c}\alpha\\\beta\end{array}\right)\rightarrow\left(\begin{array}{cc}\alpha_0&\alpha_1\\\beta_0&\beta_1\end{array}\right)$$ asserts that for every coloring $f:\alpha\times\beta\rightarrow 2$, (at least) one of the following holds: there are $A\subseteq\alpha$ and $B\subseteq\beta$ with $\text{otp}(A)=\alpha_0, \text{otp}(B)=\beta_0$ s.t. $f=\{0\}$; there ...
  • On the consistency strength of the Milner-Sauer conjecture Abstract: In their paper from 1981, Milner and Sauer conjectured that for any poset $\mathbb P$, if $\text{cf}(\mathbb P)$ is a singular cardinal $\lambda$, then $\mathbb P$ must contain an antichain of size $\text{cf}(\lambda)$. The conjecture is consistent and known to follow from GCH-type assumptions. We prove that the conjecture has large cardinals consistency strength in the ...
  • On guessing generalized clubs at the successors of regulars Abstract: Konig, Larson and Yoshinobu initiated the study of principles for guessing generalized clubs, and introduced a construction of an higher Souslin tree from the strong guessing principle. Complementary to the author’s work on the validity of diamond and non-saturation at the successor of singulars, we deal here with successor of regulars. It is established that ...
  • On topological spaces of singular density and minimal weight Abstract: We introduce a weakening of the Generalized Continuum Hypothesis, which we will refer to as the Prevalent Singular cardinals Hypothesis (PSH), and show it implies that every topological space of density and weight $\aleph_{\omega_1}$ is not hereditarily Lindelöf. The assumption PSH is very weak, and in fact holds in all currently known models of ZFC. Downloads: Citation information: A. ...
  • Polychromatic colorings These are lectures notes of two talks Dani Livne gave in our Infinite Combinatorics seminar. I did not take notes in real-time, hence, all possible mistakes here are due to myself. Recall that a function $f:A\rightarrow B$ is said to be $\theta$-to-1 if for every $b\in B$, the preimage $f^{-1}\{b\}$ has size $\theta$. It is said ...
  • Generalizations of Martin’s Axiom and the well-met condition Recall that Martin’s Axiom asserts that for every partial order $\mathbb P$ satisfying c.c.c., and for any family $\mathcal D$ of $<2^{\aleph_0}$ many dense subsets of $\mathbb P$, there exists a directed subset $G$ of $\mathbb P$ such that $G\cap D\neq\emptyset$ for all $D\in\mathcal D$. Solovay and Tennenbaum proved that Martin’s Axiom is consistent with $2^{\aleph_0}>\aleph_1$. In , ...
  • Dushnik-Miller for regular cardinals (part 1) This is the first out of a series of posts on the following theorem. Theorem (Erdos-Dushnik-Miller, 1941). For every infinite cardinal $\lambda$, we have: $$\lambda\rightarrow(\lambda,\omega)^2.$$ Namely, for any coloring $c:^2=\{1\}$. In this post, we shall focus on the case ...
  • Aspects of singular cofinality Abstract. We study properties of closure operators of singular cofinality, and introduce several ZFC sufficient and equivalent conditions for the existence of antichain sequences in posets of singular cofinality. We also notice that the Proper Forcing Axiom implies the Milner-Sauer conjecture. Downloads: Citation information: A. Rinot, Aspects of singular cofinality, Contrib. Discrete Math., 2(2): 185-204, 2007.
  • MFO workshop in Set Theory, April 2020 The Set Theory workshop in Obwerwolfach was supposed to take place April 2020, but was transformed into a webinar, due to COVID-19. Here you will find the title, abstract, slides and video of my webinar talk. Talk Title: Transformations of the transfinite plane. Abstract: We study the existence of transformations of the transfinite plane that allow to reduce ...
  • What’s next? I took an offer for a tenure-track position at the Mathematics department of Bar-Ilan University. https://www.youtube.com/watch?v=zJUXFhgN9kQ
  • 4th Arctic Set Theory Workshop, January 2019 I gave an invited talk at the Arctic Set Theory Workshop 4 in Kilpisjärvi, January 2019. Talk Title: Splitting a stationary set: Is there another way? Abstract: Motivated by a problem in pcf theory, we seek for a new way to partition stationary sets. This is joint work with Maxwell Levine. Downloads:
  • The 15th International Workshop on Set Theory in Luminy, September 2019 I gave an invited talk at the 15th International Workshop on Set Theory in Luminy in Marseille, September 2019. Talk Title: Chain conditions, unbounded colorings and the C-sequence spectrum. Abstract: The productivity of the $\kappa$-chain condition, where $\kappa$ is a regular, uncountable cardinal, has been the focus of a great deal of set-theoretic research. In the 1970’s, consistent examples ...
  • 50 Years of Set Theory in Toronto, May 2019 I gave an invited talk at the 50 Years of Set Theory in Toronto meeting, Fields Institute for Research in Mathematical Sciences, May 2019. Talk Title: Analytic quasi-orders and two forms of diamond Abstract: We study Borel reduction of equivalence relations (more generally: quasi-orders) in the generalized Baire and Cantor spaces, and connect it with the study of two ...
  • CMS Winter Meeting, December 2011 I gave an invited special session talk at the 2011 meeting of the Canadian Mathematical Society. Talk Title: The extent of the failure of Ramsey’s theorem at successor cardinals. Abstract: We shall discuss the results of the following papers: Transforming rectangles into squares, with applications to strong colorings Rectangular square-bracket operation for successor of regulars Both papers deal with strong ...
  • An inconsistent form of club guessing In this post, we shall present an answer (due to P. Larson) to a question by A. Primavesi concerning a certain strong form of club guessing. We commence with recalling Shelah’s concept of club guessing. Concept (Shelah). Given a regular uncountable cardinal $\kappa$ and a subset $S\subseteq\kappa$, we say that $\langle C_\alpha\mid\alpha\in S\rangle$ is a club-guessing sequence, ...
  • The S-space problem, and the cardinal invariant $\mathfrak p$ Recall that an $S$-space is a regular hereditarily separable topological space which is not hereditarily Lindelöf. Do they exist? Consistently, yes. However, Szentmiklóssy proved that compact $S$-spaces do not exist, assuming Martin’s Axiom. Pushing this further, Todorcevic later proved that there are no $S$-spaces under PFA. In the above-mentioned paper of Todorcevic, it is also proved ...
  • 11th Young Set Theory Workshop, June 2018 I gave a 4-lecture tutorial at the 11th Young Set Theory Workshop, Lausanne, June 2018. Title: In praise of C-sequences. Abstract. Ulam and Solovay showed that any stationary set may be split into two. Is it also the case that any fat set may be split into two? Shelah and Ben-David proved that, assuming GCH, if ...
  • The 14th International Workshop on Set Theory in Luminy, October 2017 I gave an invited talk at the 14th International Workshop on Set Theory in Luminy in Marseille, October 2017. Talk Title: Distributive Aronszajn trees Abstract: It is well-known that that the statement “all $\aleph_1$-Aronszajn trees are special” is consistent with ZFC (Baumgartner, Malitz, and Reinhardt), and even with ZFC+GCH (Jensen). In contrast, Ben-David and Shelah proved that, assuming ...
  • Variations on diamond Jensen’s diamond principle has many equivalent forms. The translation between these forms is often straight-forward, but there is one form whose equivalence to the usual form is somewhat surprising, and Devlin’s translation from one to the other, seems a little bit of a magic. Let us provide a proof. Lemma (Devlin, Kunen). The following are equivalent: There ...
  • The Engelking-Karlowicz theorem, and a useful corollary Theorem (Engelking-Karlowicz, 1965). For cardinals $\kappa\le\lambda\le\mu\le 2^\lambda$, the following are equivalent: $\lambda^{<\kappa}=\lambda$; there exists a collection of functions, $\langle f_i:\mu\rightarrow\lambda\mid i<\lambda\rangle$, such that for every $X\in^{<\kappa}$ and every function $f:X\rightarrow\lambda$, there exists some $i<\lambda$ with $f\subseteq f_i$. Proof. (2)$\Rightarrow$(1) Suppose $\langle f_i:\mu\rightarrow\lambda\mid i<\lambda\rangle$ is a given collection. Then $$|\{ f_i\restriction\theta\mid i<\lambda,\theta<\kappa\}|\le\lambda<\lambda^{<\kappa},$$ so there must exists some $f\in{}^{<\kappa}\lambda$ with $f\nsubseteq ...
  • The P-Ideal Dichotomy and the Souslin Hypothesis John Krueger is visiting Toronto these days, and in a conversation today, we asked ourselves how do one prove the Abraham-Todorcevic theorem that PID implies SH. Namely, that the next statement implies that there are no Souslin trees: Definition. The P-ideal Dichotomy asserts that for every uncountable set $Z$, and every P-Ideal $\mathcal I$ over $^{\le\aleph_0}$, ...
  • Dushnik-Miller for regular cardinals (part 3) Here is what we already know about the Dushnik-Miller theorem in the case of $\omega_1$ (given our earlier posts on the subject): $\omega_1\rightarrow(\omega_1,\omega+1)^2$ holds in ZFC; $\omega_1\rightarrow(\omega_1,\omega+2)^2$ may consistently fail; $\omega_1\rightarrow(\omega_1,\omega_1)^2$ fails in ZFC. In this post, we shall provide a proof of Todorcevic’s theorem that PFA implies $\omega_1\rightarrow(\omega_1,\alpha)^2$ for all $\alpha<\omega_1$. We commence with a sequence of lemmas. Lemma 1.  ...
  • A strong form of König’s lemma A student proposed to me the following strong form of König’s lemma: Conjecture. Suppose that $G=(V,E)$ is a countable a graph, and there is a partition of $V$ into countably many pieces $V=\bigcup_{n<\omega}V_n$, such that: for all $n<\omega$, $V_n$ is finite (but nonempty); for all two distinct $n,m<\omega$, for each $x\in V_n$, there is $y\in V_m$ with $\{x,y\}\in ...
  • 6th European Set Theory Conference, July 2017 I gave a 3-lecture tutorial at the 6th European Set Theory Conference in Budapest, July 2017. Title: Strong colorings and their applications. Abstract. Consider the following questions. Is the product of two $\kappa$-cc partial orders again $\kappa$-cc? Does there exist a regular hereditary separable topological space which is non-Lindelof? Given an $\aleph_1$-sized Abelian group $(G,+)$, must there exist a ...
  • ASL North American Meeting, March 2017 I gave a plenary talk at the 2017 ASL North American Meeting in Boise, March 2017. Talk Title: The current state of the Souslin problem. Abstract: Recall that the real line is that unique separable, dense linear ordering with no endpoints in which every bounded set has a least upper bound. A problem posed by Mikhail Souslin in 1920 asks ...
  • MFO workshop in Set Theory, February 2017 I gave an invited talk at the Set Theory workshop in Obwerwolfach, February 2017. Talk Title: Coloring vs. Chromatic. Abstract: In a joint work with Chris Lambie-Hanson, we study the interaction between compactness for the chromatic number (of graphs) and compactness for the coloring number. Downloads:
  • Set Theory and its Applications in Topology, September 2016 I gave an invited talk at the Set Theory and its Applications in Topology meeting, Oaxaca, September 11-16, 2016. The talk was on the $\aleph_2$-Souslin problem. If you are interested in seeing the effect of a jet lag, the video is available in here. Downloads:
  • Prikry forcing may add a Souslin tree A celebrated theorem of Shelah states that adding a Cohen real introduces a Souslin tree. Are there any other examples of notions of forcing that add a $\kappa$-Souslin tree? and why is this of interest? My motivation comes from a question of Schimmerling, which I shall now motivate and state. Recall that Jensen proved that GCH together ...
  • The reflection principle $R_2$ A few years ago, in this paper, I introduced the following reflection principle: Definition. $R_2(\theta,\kappa)$ asserts that for every function $f:E^\theta_{<\kappa}\rightarrow\kappa$, there exists some $j<\kappa$ for which the following set is nonstationary: $$A_j:=\{\delta\in E^\theta_\kappa\mid f^{-1}\cap\delta\text{ is nonstationary}\}.$$ I wrote there that by a theorem of Magidor, $R_2(\aleph_2,\aleph_1)$ is consistent modulo the existence of a weakly compact cardinal, ...
  • Prolific Souslin trees In a paper from 1971, Erdos and Hajnal asked whether (assuming CH) every coloring witnessing $\aleph_1\nrightarrow^2_3$ has a rainbow triangle. The negative solution was given in a 1975 paper by Shelah, and the proof and relevant definitions may be found in this previous blog post. Shelah’s construction of a coloring witnessing $\aleph_1\nrightarrow^2_\omega$ with no rainbow triangles ...
  • P.O.I. Workshop in pure and descriptive set theory, September 2015 I gave an invited talk at the P.O.I Workshop in pure and descriptive set theory, Torino, September 26, 2015.  Title: $\aleph_3$-trees. Abstract: We inspect the constructions of four quite different $\aleph_3$-Souslin trees. Downloads:
  • The Apter-Gitik birthday conference, May 2015 I give an invited (blackboard) talk at the Apter-Gitik birthday conference, Carnegie Mellon University, May 30-31 2015.  Title: Putting a diamond inside the square. Abstract: By a 35-year-old theorem of Shelah, $\square_\lambda+\diamondsuit(\lambda^+)$ does not imply square-with-built-in-diamond_lambda for regular uncountable cardinals $\lambda$. In this talk, I will sketch the proof from my recent paper, stating that $\square_\lambda+\diamondsuit(\lambda^+)$ is equivalent ...
  • Forcing and its Applications Retrospective Workshop, April 2015 I gave an invited talk at Forcing and its Applications Retrospective Workshop, Toronto, April 1st, 2015.  Title: A microscopic approach to Souslin trees constructions Abstract: We present an approach to construct $\kappa$-Souslin trees that is insensitive to the identity of the cardinal $\kappa$, thereby, allowing to transform constructions from successor of regulars to successor of singulars and ...
  • Many diamonds from just one Recall Jensen’s diamond principle over a stationary subset $S$ of a regular uncountable cardinal $\kappa$: there exists a sequence $\langle A_\alpha\mid \alpha\in S \rangle$ such that $\{\alpha\in S\mid A\cap\alpha=A_\alpha\}$ is stationary for every $A\subseteq\kappa$. Equivalently, there exists a sequence $\langle f_\alpha:\alpha\rightarrow\alpha\mid \alpha\in S\rangle$ such that $\{\alpha\in S\mid f\restriction\alpha=f_\alpha\}$ is stationary for every function $f:\kappa\rightarrow\kappa$. It is ...
  • Happy new jewish year!
  • INFTY Final Conference, March 2014 I gave an invited talk at the INFTY Final Conference meeting, Bonn, March 4-7, 2014. Title: Same Graph, Different Universe. Abstract: In a paper from 1998, answering a question of Hajnal, Soukup proved that ZFC+GCH is consistent with the existence of two graphs G,H of size and chromatic number ...
  • Partitioning the club guessing In a recent paper, I am making use of the following  fact. Theorem (Shelah, 1997). Suppose that $\kappa$ is an accessible cardinal (i.e., there exists a cardinal $\theta<\kappa$ such that $2^\theta\ge\kappa)$. Then there exists a sequence $\langle g_\delta:C_\delta\rightarrow\omega\mid \delta\in E^{\kappa^+}_\kappa\rangle$ such that $C_\delta$ is a club of $\delta$ of order-type $\kappa$, and with the property that ...
  • MFO workshop in Set Theory, January 2014 I gave an invited talk at the Set Theory workshop in Obwerwolfach, January 2014. Talk Title: Complicated Colorings. Abstract: If $\lambda,\kappa$ are regular cardinals, $\lambda>\kappa^+$, and $E^{\lambda}_{\ge\kappa}$ admits a nonreflecting stationary set, then $\text{Pr}_1(\lambda,\lambda,\lambda,\kappa)$ holds. Downloads:
  • Walk on countable ordinals: the characteristics In this post, we shall present a few aspects of the method of walk on ordinals (focusing on countable ordinals), record its characteristics, and verify some of their properties. All definitions and results in this post are due to Todorcevic. Let $\langle C_\alpha\mid\alpha<\omega_1\rangle$ be a sequence such that $C_{\alpha+1}=\{\alpha\}$ for all $\alpha<\omega_1$, and for all limit ...
  • Universal binary sequences Notation. Write $\mathcal Q(A):=\{ a\subseteq A\mid a\text{ is finite}, a\neq\emptyset\}$. Suppose for the moment that we are given a fixed sequence $\langle f_\alpha:\omega\rightarrow2\mid \alpha\in a\rangle$, indexed by some set $a$ of ordinals. Then, for every function $h:a\rightarrow\omega$ and $i<\omega$, we define the $i$-valuation of $h$, $h^i:a\rightarrow2$, by letting for all $\alpha\in a$: $$h^i(\alpha):=f_\alpha(h(\alpha)+i).$$ Given an initial state $h:a\rightarrow\omega$, ...
  • Syndetic colorings with applications to S and L Notation. Write $\mathcal Q(A):=\{ a\subseteq A\mid a\text{ is finite}, a\neq\emptyset\}$. Definition. An L-space is a regular hereditarily Lindelöf topological space which is not hereditarily separable. Definition. We say that a coloring $c:^2\rightarrow\omega$ is L-syndetic if the following holds. For every uncountable family $\mathcal A\subseteq\mathcal Q(\omega_1)$ of mutually disjoint sets, every uncountable $B\subseteq\omega_1$, and every $n<\omega$, there exist $a\in\mathcal A$, ...
  • Open coloring and the cardinal invariant $\mathfrak b$ Nik Weaver asked for a direct proof of the fact that Todorcevic’s axiom implies the failure of CH fails. Here goes. Notation. For a set $X$, we write $^2$ for the set of unordered pairs $\{ \{x,x’\}\mid x,x’\in X, x\neq x’\}$. Definition (Todorcevic’s axiom). If $X$ is a separable metric space, and $^2=K_0\cup K_1$, with $K_0$ open ...
  • Gabriel Belachsan (14/5/1976 – 20/8/2013) רק כשעיני סגורות, עולם נגלה לפני http://www.youtube.com/watch?v=6Mq-xMhhe3s
  • Hedetniemi’s conjecture for uncountable graphs Abstract. It is proved that in Godel’s constructible universe, for every successor cardinal $\kappa$, there exist graphs $\mathcal G$ and $\mathcal H$ of size and chromatic number $\kappa$, for which the tensor product graph $\mathcal G\times\mathcal H$ is countably chromatic. Downloads: Citation information: A. Rinot, Hedetniemi’s conjecture for uncountable graphs, J. Eur. Math. Soc., 19(1): 285-298, 2017.
  • Set Theory Programme on Large Cardinals and Forcing, September 2013 I gave an invited talk at the Large Cardinals and Forcing meeting, Erwin Schrödinger International Institute for Mathematical Physics, Vienna, September 23–27, 2013. Talk Title: Hedetniemi’s conjecture for uncountable graphs Abstract: It is proved that in Godel’s constructible universe, for every successor cardinal $\kappa$, there exist graphs $\mathcal G$ and $\mathcal H$ of size and chromatic number ...
  • PFA and the tree property at $\aleph_2$ Recall that a poset $\langle T,\le\rangle$ is said to be a $\lambda^+$-Aronszajn tree, if it isomorphic to a poset $(\mathcal T,\subseteq)$ of the form: $\emptyset\in \mathcal T\subseteq{}^{<\lambda^+}\lambda$; Write $\mathcal T_\alpha:=\{\sigma\in\mathcal T\mid \text{dom}(\sigma)=\alpha\}$; for all $\alpha<\lambda^+$, $\mathcal T_\alpha$ has size $\le\lambda$, say $\mathcal T_\alpha=\{ T_\alpha^i\mid i<\lambda\}$; if $\sigma\in\mathcal T$ and $\alpha<\lambda^+$, then there exists $\tau\in\mathcal T_\alpha$ such that $\sigma\cup\tau\in{}^{<\lambda^+}\lambda$; $\mathcal ...
  • A Kurepa tree from diamond-plus Recall that $T$ is said to be a $\kappa$-Kurepa tree if $T$ is a tree of height $\kappa$, whose levels $T_\alpha$ has size $\le|\alpha|$ for co-boundedly many $\alpha<\kappa$, and such that the set of branches of $T$ has size $>\kappa$. In this post, we shall remind ourselves of the proof of Jensen’s theorem that $\diamondsuit^+(\kappa)$ entails ...
  • The S-space problem, and the cardinal invariant $\mathfrak b$ Recall that an S-space is a regular hereditarily separable topological space which is not hereditarily Lindelöf. In a previous post, we showed that such a space exists after adding a Cohen real. Here, we shall construct one from an arithmetic assumption. Theorem (Todorcevic, 1989). If $\mathfrak b=\omega_1$, then there exists an $S$-space. Proof. Let $\overrightarrow f=\langle f_\alpha\mid\alpha<\omega_1\rangle$ ...
  • The S-space problem, and the cardinal invariant $\mathfrak b$ Recall that an S-space is a regular hereditarily separable topological space which is not hereditarily Lindelöf. In a previous post, we showed that such a space exists after adding a Cohen real. Here, we shall construct one from an arithmetic assumption. Theorem (Todorcevic, 1989). If $\mathfrak b=\omega_1$, then there exists an $S$-space. Proof. Let $\overrightarrow f=\langle f_\alpha\mid\alpha<\omega_1\rangle$ ...
  • An $S$-space from a Cohen real Recall that an $S$-space is a regular hereditarily separable topological space which is not hereditarily Lindelöf. In this post, we shall establish the consistency of the existence of such a space. Theorem (Roitman, 1979). Let $\mathbb C=({}^{<\omega}\omega,\subseteq)$ be the notion of forcing for adding a Cohen real. Then, in the generic extension by forcing with $\mathbb C$, ...
  • Forcing with a Souslin tree makes $\mathfrak p=\omega_1$ I was meaning to include a proof of Farah’s lemma in my previous post, but then I realized that the slick proof assumes some background which may worth spelling out, first. Therefore, I am dedicating a short post for a self-contained proof of this lemma. Recall that a Souslin tree is a poset $\mathbb T=\langle T,<\rangle$ ...
  • Forcing with a Souslin tree makes $\mathfrak p=\omega_1$ I was meaning to include a proof of Farah’s lemma in my previous post, but then I realized that the slick proof assumes some background which may worth spelling out, first. Therefore, I am dedicating a short post for a self-contained proof of this lemma. Recall that a Souslin tree is a poset $\mathbb T=\langle T,<\rangle$ ...
  • Jones’ theorem on the cardinal invariant $\mathfrak p$ This post continues the study of the cardinal invariant $\mathfrak p$. We refer the reader to a previous post for all the needed background. For ordinals $\alpha,\alpha_0,\alpha_1,\beta,\beta_0,\beta_1$, the polarized partition relation $$\left(\begin{array}{c}\alpha\\\beta\end{array}\right)\rightarrow\left(\begin{array}{cc}\alpha_0&\alpha_1\\\beta_0&\beta_1\end{array}\right)$$ asserts that for every coloring $f:\alpha\times\beta\rightarrow 2$, (at least) one of the following holds: there are $A\subseteq\alpha$ and $B\subseteq\beta$ with $\text{otp}(A)=\alpha_0, \text{otp}(B)=\beta_0$ s.t. $f=\{0\}$; there ...
  • Erdős 100 The influential mathematician Paul Erdős was born 100 years ago, 26 March 1913, in Budapest.  One evidence of his impact on mathematics is reflected in the particular list of invited speakers for the upcoming conference in his honor. Erdős is also famous for taking his birthdays somewhat seriously. Here is a funny video of him talking about ...
  • Bell’s theorem on the cardinal invariant $\mathfrak p$ In this post, we shall provide a proof to a famous theorem of Murray Bell stating that $MA_\kappa(\text{the class of }\sigma\text{-centered posets})$ holds iff $\kappa<\mathfrak p$. We commence with defining the cardinal invariant $\mathfrak p$. For sets $A$ and $B$, we write $B\subseteq^* A$ iff $\sup(B\setminus A)<\sup(B)$. We say that $B$ is a pseudointersection of a ...
  • Bell’s theorem on the cardinal invariant $\mathfrak p$ In this post, we shall provide a proof to a famous theorem of Murray Bell stating that $MA_\kappa(\text{the class of }\sigma\text{-centered posets})$ holds iff $\kappa<\mathfrak p$. We commence with defining the cardinal invariant $\mathfrak p$. For sets $A$ and $B$, we write $B\subseteq^* A$ iff $\sup(B\setminus A)<\sup(B)$. We say that $B$ is a pseudointersection of a ...
  • The $\Delta$-system lemma: an elementary proof Here is an elementary proof of (the finitary version of) the $\Delta$-system lemma. Thanks goes to Bill Weiss who showed me this proof! Lemma. Suppose that $\kappa$ is a regular uncountable cardinal, and $\mathcal A$ is a $\kappa$-sized family of finite sets. Then there exists a subcollection $\mathcal B\subseteq\mathcal A$ of size $\kappa$, together with a finite set ...
  • A natural Mandelbrot set Chris Hadfield is a Canadian astronaut, with a very high-profile twitter account. He posts there beautiful photos everyday, and I (plus half a million followers) enjoy it very much. Today, Chris posted the following picture: and I find it quite similar to the famous Mandelbrot set: Recall that the so-called Mandelbrot set is a fractal whose shape consists ...
  • Prikry Forcing Recall that the chromatic number of a (symmetric) graph $(G,E)$, denoted $\text{Chr}(G,E)$, is the least (possible finite) cardinal $\kappa$, for which there exists a coloring $c:G\rightarrow\kappa$ such that $gEh$ entails $c(g)\neq c(h)$. Given a forcing notion $\mathbb P$, it is natural to analyze $\text{Chr}(\mathbb P,\bot)$, where $p\bot q$ iff $p$ and $q$ are incompatible conditions. Here ...
  • The uniformization property for $\aleph_2$ Given a subset of a regular uncountable cardinal $S\subseteq\kappa$, $UP_S$ (read: “the uniformization property holds for $S$”) asserts that for every sequence $\overrightarrow f=\langle f_\alpha\mid \alpha\in S\rangle$ satisfying for all $\alpha\in S$: $f_\alpha$ is a 2-valued function; $\text{dom}(f_\alpha)$ is a cofinal subset of $\alpha$ of minimal order-type, there exists a function $f:\kappa\rightarrow 2$ such that for every limit ...
  • The uniformization property for $\aleph_2$ Given a subset of a regular uncountable cardinal $S\subseteq\kappa$, $UP_S$ (read: “the uniformization property holds for $S$”) asserts that for every sequence $\overrightarrow f=\langle f_\alpha\mid \alpha\in S\rangle$ satisfying for all $\alpha\in S$: $f_\alpha$ is a 2-valued function; $\text{dom}(f_\alpha)$ is a cofinal subset of $\alpha$ of minimal order-type, there exists a function $f:\kappa\rightarrow 2$ such that for every limit ...
  • Kurepa trees and ineffable cardinals Recall that $T$ is said to be a $\kappa$-Kurepa tree if $T$ is a tree of height $\kappa$, whose levels $T_\alpha$ has size $\le|\alpha|$ for co-boundedly many $\alpha<\kappa$, and such that the set of branches of $T$ has size $>\kappa$. Recall also that an uncountable cardinal $\kappa$ is said to be ineffable if for every sequence ...
  • Infinite Combinatorics Seminar, Haifa University, June, 2012 I gave a talk at the University of Haifa on June 07, 2012, intended for general audience. Title: Strong Colorings: the study of the failure of generalized Ramsey statements Abstract: A strong coloring from $X$ to $Y$ is a function that transforms relatively thin subsets of $X$ into relatively fat subsets of $Y$. The first example of ...
  • Afghan Whigs on Jimmy Fallon Performing “I’m Her Slave” (from their album Congregation) at NBC’s studios, 22-May-2012:
  • Shelah’s approachability ideal (part 1) Given an infinite cardinal $\lambda$, Shelah defines an ideal $I$ as follows. Definition (Shelah, implicit in here). A set $S$ is in $I^{<\lambda}$, and some club $E\subseteq\lambda$, so that for every $\delta\in S\cap E$, there exists a cofinal subset $A_\delta\subseteq\delta$ such that: $\text{otp}(A_\delta)<\delta$ (in particular, ...
  • Review: Is classical set theory compatible with quantum experiments? Yesterday, I attended a talk at the Quantum Foundations seminar at the beautiful Perimeter Institute for Theoretical Physics (Waterloo, Ontario). The (somewhat provocative) title of the talk was “Is Classical Set Theory Compatible with Quantum Experiments?”, and the speaker was Radu Ionicioiu. Here are the links to the slides and videotape. To make a long story short, the ...
  • Comparing rectangles with squares through rainbow sets In Todorcevic’s class last week, he proved all the results of Chapter 8 from his Walks on Ordinals book, up to (and including) Theorem 8.1.11. The upshots are as follows: Every regular infinite cardinal $\theta$ admits a naturally defined function $osc:^2\rightarrow\omega$; there exists, again natural, notion of unbounded subfamily of $\mathcal P(\theta)$, and if $\mathcal X$ ...
  • ASL North American Meeting, March 2012 I gave a special session talk at the ASL 2012 North American Annual Meeting (Madison, March 31–April 3, 2012). Talk Title: The extent of the failure of Ramsey’s theorem at successor cardinals. Extended abstract: Ramsey’s theorem asserts that for every coloring $c:^2$ is constant. At the early 1930’s, ...
  • Pure logic While traveling downtown today, I came across a sign near a local church, with a quotation of Saint-Exupéry: “Pure logic is the ruin of the spirit”? So sad.
  • Jane’s Addiction visiting Toronto Last night, I went to see a live show by Jane’s Addiction, in downtown Toronto. Here’s a video snippet from that show which I could found on YouTube: http://www.youtube.com/watch?v=vprun9x8QWE The playlist was excellent, but there was one song which I was missing – “Had a dad”, so I compensate the loss, by including it here..: http://www.youtube.com/watch?v=zdPCwxCg1a4
  • c.c.c. vs. the Knaster property After my previous post on Mekler’s characterization of c.c.c. notions of forcing, Sam, Mike and myself discussed the value of it . We noticed that a prevalent verification of the c.c.c. goes like this: given an uncountable set of conditions, apply the $\Delta$-system lemma, thin out the outcome some more (typically, a few iterations are ...
  • A large cardinal in the constructible universe In this post, we shall provide a proof of Silver’s theorem that the Erdos caridnal $\kappa(\omega)$ relativizes to Godel’s constructible universe. First, recall some definitions. Given a function $f:^n$, we have $f(x)=f(y)$. Let $\kappa\rightarrow(\alpha)^{<\omega}_\mu$ denote the ...
  • c.c.c. forcing without combinatorics In this post, we shall discuss a short paper by Alan Mekler from 1984, concerning a non-combinatorial verification of the c.c.c. property for forcing notions. Recall that a notion of forcing $\mathbb P$ is said to satisfy the c.c.c. iff all of its antichains are countable. A generalization of this property is that of being proper: Definition ...
  • Dushnik-Miller for singular cardinals (part 2) In the first post on this subject, we provided a proof of $\lambda\rightarrow(\lambda,\omega+1)^2$ for every regular uncountable cardinal $\lambda$. In the second post, we provided a proof of $\lambda\rightarrow(\lambda,\omega)^2$ for every singular cardinal $\lambda$, and showed that $\lambda\rightarrow(\lambda,\omega+1)^2$ fails for every cardinal of countable cofinality. In this post, we shall be dealing with the missing case, proving ...
  • Dushnik-Miller for regular cardinals (part 2) In this post, we shall provide a proof of Todorcevic’s theorem, that $\mathfrak b=\omega_1$ implies $\omega_1\not\rightarrow(\omega_1,\omega+2)^2$. This will show that the Erdos-Rado theorem that we discussed in an earlier post, is consistently optimal. Our exposition of Todorcevic’s theorem would be slightly more general, with the benefit of yielding another famous theorem of Todorcevic: $\omega_1\not\rightarrow^2_{\omega_1}$. We commence ...
  • Dushnik-Miller for singular cardinals (part 1) Continuing the previous post, let us now prove the following. Theorem (Erdos-Dushnik-Miller, 1941). For every singular cardinal λ, we have: $$\lambda\rightarrow(\lambda,\omega)^2.$$ Proof. Suppose that $\lambda$ is a singular cardinal, and $c:^2\rightarrow\{0,1\}$ is a given coloring. For any ordinal $\alpha<\lambda$, denote $I_\alpha:=\{ \beta\in(\alpha,\lambda)\mid c(\alpha,\beta)=1\}$.   Subclaim. At least one of the following holds: there exists an infinite subset $B\subseteq\lambda$ for which $c“^2=\{1\}$; there ...
  • The order-type of clubs in a square sequence Recall Jensen’s notion of square: Definition (Jensen): For an infinite cardinal $\lambda$, $\square_\lambda$ asserts the existence of a sequence $\overrightarrow C=\left\langle C_\alpha\mid\alpha\in\text{acc}(\lambda^+)\right\rangle$ such that for every limit $\alpha<\lambda^+$: $C_\alpha$ is a club subset of $\alpha$ of order-type $\le\lambda$; if $\beta\in\text{acc}(C_\alpha)$, then $C_\beta=C_\alpha\cap\beta$. Now, consider the set $S(\overrightarrow C):=\{ \alpha<\lambda^+\mid \text{otp}(C_\alpha)=\lambda\}$. If $\lambda$ is a regular cardinal, then $S(\overrightarrow C)=\{\alpha<\lambda^+\mid\text{cf}(\alpha)=\lambda\}$, and ...
  • James Earl Baumgartner Sad news: Jim Baumgartner passed away. See here.
  • The search for diamonds Abstract: This is a review I wrote for the Bulletin of Symbolic Logic  on the following papers: Saharon Shelah, Middle Diamond, Archive for Mathematical Logic, vol. 44 (2005), pp. 527–560. Saharon Shelah, Diamonds, Proceedings of the American Mathematical Society, vol. 138 no. 6 (2010), pp. 2151–2161. Martin Zeman. Diamond, GCH and Weak Square. Proceedings of the American Mathematical ...
  • A cofinality-preserving small forcing may introduce a special Aronszajn tree Extended Abstract: Shelah proved that Cohen forcing introduces a Souslin tree; Jensen proved that a c.c.c. forcing may consistently add a Kurepa tree; Todorcevic proved that a Knaster poset may already force the Kurepa hypothesis; Irrgang introduced a c.c.c. notion of forcing based on a simplified ($\omega_1$,1)-morass that adds an $\omega_2$-Souslin tree. Here, it is proved that adding a subset ...
  • Young Researchers in Set Theory, March 2011 These are the slides of a talk I gave at the Young Researchers in Set Theory 2011 meeting (Königswinter, 21–25 March 2011). Talk Title: Around Jensen’s square principle Abstract: Jensen‘s square principle for a cardinal $\lambda$ asserts the existence of a particular ladder system over $\lambda^+$. This principle admits a long list of applications including the existence of non-reflecting stationary sets, ...
  • The 11th International Workshop on Set Theory in Luminy The 11th International Workshop on Set Theory in Luminy
  • Logic Colloquium 2010 Logic Colloquium 2010
  • Logic Colloquium 2009 Logic Colloquium 2009
  • ESI Workshop on Large Cardinals and Descriptive Set Theory ESI Workshop on Large Cardinals and Descriptive Set Theory
  • 18th Boise Extravaganza in Set Theory 18th Boise Extravaganza in Set Theory
  • Singular Cardinal Combinatorics and Inner Model Theory, March 2007 These are the slides of a talk given at the Singular Cardinal Combinatorics and Inner Model Theory conference (Gainesville, 5–9 March 2007). Talk Title: Antichains in partially ordered sets of singular cofinality Abstract: We say that a singular cardinal $\lambda$ is a prevalent singular cardinal iff there exists a family $\mathcal{F}$ of size $\lambda$ with $\sup\{ |A| : A\in\mathcal{F}\}<\lambda$ such that any subset of $\lambda$ ...
  • Workshop on Set Theory and its Applications, February 2007 These are the slides of a talk given at the Workshop on Set Theory and its Applications workshop (Weizmann Institute, February 19, 2007). Talk Title: Nets of spaces having singular density Abstract: The weight of a topological space X is the minimal cardinality of basis B for X. The density of X is the minimal cardinality of a ...
  • Antichains in partially ordered sets of singular cofinality Abstract: In their paper from 1981, Milner and Sauer conjectured that for any poset $\mathbb P$, if $\text{cf}(\mathbb P)$ is a singular cardinal $\lambda$, then $\mathbb P$ must contain an antichain of size $\text{cf}(\lambda)$. The main result of of this paper reads as follows: Suppose that $\lambda$ singular cardinal. If there exists a cardinal $\mu<\lambda$ for which $\text{cov}\left(\lambda,\mu,\text{cf}(\lambda),2\right)=\lambda$, ...
  • The Ostaszewski square, and homogeneous Souslin trees Abstract: Assume GCH and let $\lambda$ denote an uncountable cardinal. We prove that if $\square_\lambda$ holds, then this may be  witnessed by a coherent sequence $\left\langle C_\alpha \mid \alpha<\lambda^+\right\rangle$ with the following remarkable guessing property: For every sequence $\langle A_i\mid i<\lambda\rangle$ of unbounded subsets of $\lambda^+$, and every limit $\theta<\lambda$, there exists some $\alpha<\lambda^+$ such that $\text{otp}(C_\alpha)=\theta$, and ...
  • Infinite Combinatorial Topology Back in 2005, as a master student, I attended a course by Boaz Tsaban, entitled “Infinite Combinatorial Topology”. A friend and I decided to produce lecture notes, but in a somewhat loose sense, that is: we sometimes omit material given in class, sometimes give alternative definitions or proofs, and sometimes include our own additional propositions. ...