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