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