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Minimal Walks Strongly compact cardinal weak square Antichain Coherent tree Filter reflection specializable Souslin tree Local Club Condensation. Well-behaved magma stationary reflection HOD Square-Brackets Partition Relations transformations Erdos-Hajnal graphs O-space approachability ideal Weakly compact cardinal Luzin set Cohen real Cardinal Invariants PFA Partition Relations Intersection model Amenable C-sequence Almost countably chromatic stationary hitting Knaster and friends Closed coloring Forcing with side conditions Parameterized proxy principle Foundations Entangled linear order perfectly normal Fat stationary set Nonspecial tree Selective Ultrafilter Ostaszewski square Uniformly coherent Souslin Tree Mandelbrot set Sierpinski's onto mapping principle reflection principles Respecting tree diamond star S-Space Postprocessing function square Whitehead Problem Erdos Cardinal nonmeager set Kurepa Hypothesis Subadditive Poset Sakurai's Bell inequality Open Access L-space Slim tree Rainbow sets strongly bounded groups weak diamond Distributive tree free Souslin tree GMA projective Boolean algebra Uniformly homogeneous sap ccc 54G20 Large Cardinals Forcing Axioms Martin's Axiom b-scale super-Souslin tree unbounded function countably metacompact Cardinal function Greatly Mahlo Club Guessing Fast club Ulam matrix Countryman line AIM forcing Small forcing square principles positive partition relation weak Kurepa tree Diamond-sharp Axiom R Reflecting stationary set tensor product graph Chang's conjecture Subnormal ideal Diamond C-sequence Analytic sets Singular cofinality Ascending path Rado's conjecture higher Baire space Vanishing levels Interval topology on trees Universal Sequences Diamond for trees Lipschitz reduction Jonsson cardinal xbox OCA Singular cardinals combinatorics middle diamond Almost Souslin Hedetniemi's conjecture Precaliber Commutative projection system Reduced Power Subtle tree property Rock n' Roll Partition relations for trees Generalized descriptive set theory Singular Density Hereditarily Lindelöf space Knaster free Boolean algebra Iterated forcing Ramsey theory over partitions P-Ideal Dichotomy Absoluteness Shelah's Strong Hypothesis Commutative cancellative semigroups Constructible Universe full tree Fodor-type reflection Forcing Strongly Luzin set Successor of Regular Cardinal PFA(S)[S] Microscopic Approach Generalized Clubs Uniformization Hindman's Theorem Dowker space indecomposable filter Was Ulam right? Strong coloring Non-saturation regressive Souslin tree Ineffable cardinal stick polarized partition relation incompactness Prevalent singular cardinals Ascent Path Aronszajn tree coloring number Chromatic number Prikry-type forcing Dushnik-Miller very good scale Almost-disjoint family Subtle cardinal ZFC construction Successor of Singular Cardinal Monotonically far SNR club_AD Sigma-Prikry
Category Archives: Open Problems
May the successor of a singular cardinal be Jonsson?
Abstract: Whether the successor of a singular cardinal can be Jónsson is a very old and famous open problem in set theory. Here, we collect necessary conditions for an affirmative answer, and put forward a list of closely-related questions in … Continue reading
Posted in Open Problems, Singular Cardinals Combinatorics
Tagged Jonsson cardinal, Open Access
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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
Comments Off on Perspectives on Set Theory, November 2023
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
4 Comments
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
