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