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Was Ulam right Erdos-Hajnal graphs Prikry-type forcing square principles Cohen real free Souslin tree b-scale projective Boolean algebra Non-saturation Subadditive stationary reflection Hedetniemi's conjecture Almost countably chromatic Club Guessing Weakly compact cardinal P-Ideal Dichotomy Minimal Walks Rock n' Roll tensor product graph Commutative cancellative semigroups Ascent Path Chang's conjecture Shelah's Strong Hypothesis OCA Sigma-Prikry Nonspecial tree PFA S-Space Souslin Tree Local Club Condensation. GMA Uniformly homogeneous Ramsey theory over partitions countably metacompact HOD strongly bounded groups Axiom R Absoluteness Diamond for trees approachability ideal Aronszajn tree Jonsson cardinal Slim tree indecomposable ultrafilter Almost Souslin Forcing Axioms Open Access Dowker space Foundations Intersection model Large Cardinals Erdos Cardinal Subtle cardinal Uniformly coherent Well-behaved magma Countryman line ZFC construction weak diamond Luzin set Cardinal function Poset Microscopic Approach Hindman's Theorem Ineffable cardinal Prevalent singular cardinals Generalized descriptive set theory Knaster and friends middle diamond Subtle tree property Constructible Universe C-sequence Filter reflection Precaliber coloring number free Boolean algebra Closed coloring Analytic sets Mandelbrot set Rado's conjecture Chromatic number positive partition relation specializable Souslin tree Uniformization Martin's Axiom O-space Iterated forcing full tree Amenable C-sequence Small forcing Whitehead Problem Subnormal ideal Partition Relations L-space super-Souslin tree Fat stationary set weak Kurepa tree square Forcing stick Almost-disjoint family regressive Souslin tree Knaster Antichain weak square Postprocessing function stationary hitting Reduced Power Singular Density Respecting tree polarized partition relation Vanishing levels xbox reflection principles Parameterized proxy principle Strong coloring PFA(S)[S] transformations Distributive tree Generalized Clubs diamond star higher Baire space Kurepa Hypothesis Successor of Regular Cardinal Strongly Luzin set incompactness unbounded function Singular cardinals combinatorics club_AD Fodor-type reflection Diamond Hereditarily Lindelöf space Cardinal Invariants Reflecting stationary set Fast club sap Strongly compact cardinal Rainbow sets ccc nonmeager set Ulam matrix Singular cofinality Commutative projection system Ostaszewski square 54G20 Diamond-sharp very good scale Successor of Singular Cardinal Dushnik-Miller Coherent tree Lipschitz reduction SNR Selective Ultrafilter Greatly Mahlo Universal Sequences Square-Brackets Partition Relations Sakurai's Bell inequality Sierpinski's onto mapping principle AIM forcing
Category Archives: Open Problems
May the successor of a singular cardinal be Jonsson?
Abstract: We collect necessary conditions for the successor of a singular cardinal to be Jónsson.
Posted in Open Problems, Singular Cardinals Combinatorics
Tagged Jonsson cardinal
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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 … Continue reading
Posted in Invited Talks, Open Problems
Tagged Jonsson cardinal, Successor of Singular Cardinal
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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 … Continue reading
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 … Continue reading
Posted in Invited Talks, Open Problems
Tagged b-scale, Cohen real, Luzin set, Minimal Walks, Souslin Tree, Square-Brackets Partition Relations
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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 … Continue reading
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 $\langle g_\delta:C_\delta\rightarrow\omega\mid \delta\in E^{\kappa^+}_\kappa\rangle$ … Continue reading
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 $c:[\omega_1]^2\rightarrow\omega$ is L-syndetic if the following holds. For every uncountable … Continue reading
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 … Continue reading
Posted in Blog, Expository, Open Problems
Tagged Hereditarily Lindelöf space, P-Ideal Dichotomy, PFA(S)[S], S-Space
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Shelah’s approachability ideal (part 2)
In a previous post, we defined Shelah’s approachability ideal $I[\lambda]$. We remind the reader that a subset $S\subseteq\lambda$ is in $I[\lambda]$ iff there exists a collection $\{ \mathcal D_\alpha\mid\alpha<\lambda\}\subseteq\mathcal [\mathcal P(\lambda)]^{<\lambda}$ such that for club many $\delta\in S$, the union … Continue reading
Posted in Blog, Expository, Open Problems
Tagged approachability ideal, Club Guessing
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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 … Continue reading
