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Sakurai's Bell inequality full tree Generalized Clubs Almost Souslin b-scale polarized partition relation unbounded function Luzin set indecomposable ultrafilter Knaster and friends Slim tree Souslin Tree Fat stationary set Singular Density Chang's conjecture stationary hitting Foundations Precaliber incompactness Club Guessing Successor of Singular Cardinal Subnormal ideal Sigma-Prikry P-Ideal Dichotomy Cardinal Invariants Jonsson cardinal Was Ulam right Almost-disjoint family free Boolean algebra Postprocessing function PFA L-space Weakly compact cardinal Constructible Universe strongly bounded groups Vanishing levels Subtle tree property reflection principles nonmeager set Rainbow sets Poset Universal Sequences club_AD Ramsey theory over partitions AIM forcing Fodor-type reflection Nonspecial tree Non-saturation Diamond-sharp Dowker space C-sequence Iterated forcing Well-behaved magma Ostaszewski square Ineffable cardinal S-Space Successor of Regular Cardinal stick Whitehead Problem Shelah's Strong Hypothesis Large Cardinals ccc Uniformization Ascent Path Diamond for trees Analytic sets weak diamond Singular cardinals combinatorics tensor product graph projective Boolean algebra Absoluteness Subtle cardinal Almost countably chromatic Square-Brackets Partition Relations Mandelbrot set sap Kurepa Hypothesis middle diamond Distributive tree Uniformly coherent Diamond Chromatic number Prikry-type forcing Hindman's Theorem 54G20 Greatly Mahlo Dushnik-Miller Antichain square xbox higher Baire space Strong coloring Knaster Microscopic Approach Parameterized proxy principle Ulam matrix countably metacompact diamond star Small forcing Hereditarily Lindelöf space Sierpinski's onto mapping principle Fast club Selective Ultrafilter regressive Souslin tree free Souslin tree transformations SNR Erdos Cardinal Singular cofinality coloring number Uniformly homogeneous ZFC construction Strongly Luzin set Closed coloring HOD Local Club Condensation. Amenable C-sequence Forcing Axioms approachability ideal O-space OCA Martin's Axiom Filter reflection Rado's conjecture Commutative cancellative semigroups Rock n' Roll Reflecting stationary set GMA very good scale positive partition relation Aronszajn tree Reduced Power Open Access PFA(S)[S] Generalized descriptive set theory specializable Souslin tree Lipschitz reduction Hedetniemi's conjecture Cardinal function Coherent tree Prevalent singular cardinals Minimal Walks Forcing Partition Relations weak square square principles super-Souslin tree Erdos-Hajnal graphs Axiom R Cohen real Subadditive stationary reflection
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, Partition Relations
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