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Diamond for trees Foundations xbox Uniformly coherent Minimal Walks Cardinal function Jonsson cardinal free Boolean algebra Was Ulam right? middle diamond OCA unbounded function Antichain Strongly compact cardinal Aronszajn tree Prevalent singular cardinals very good scale Sigma-Prikry Rainbow sets transformations incompactness Ramsey theory over partitions Filter reflection Dushnik-Miller Ostaszewski square Non-saturation AIM forcing polarized partition relation Kurepa Hypothesis Cardinal Invariants free Souslin tree regressive Souslin tree Forcing weak square GMA Square-Brackets Partition Relations nonmeager set Knaster and friends diamond star Reduced Power Ascent Path Almost-disjoint family Axiom R Countryman line Absoluteness Almost countably chromatic Club Guessing Singular Density reflection principles Almost Souslin indecomposable ultrafilter Sierpinski's onto mapping principle weak diamond Parameterized proxy principle projective Boolean algebra Distributive tree Dowker space Reflecting stationary set Coherent tree PFA(S)[S] Uniformization Erdos Cardinal Shelah's Strong Hypothesis countably metacompact square principles ccc stationary reflection tensor product graph Well-behaved magma Singular cofinality b-scale Fast club P-Ideal Dichotomy stick Universal Sequences Luzin set approachability ideal Successor of Singular Cardinal Iterated forcing Whitehead Problem Amenable C-sequence positive partition relation Knaster super-Souslin tree specializable Souslin tree Selective Ultrafilter Nonspecial tree weak Kurepa tree higher Baire space Souslin Tree Rock n' Roll club_AD Open Access C-sequence Successor of Regular Cardinal Vanishing levels Rado's conjecture L-space Generalized descriptive set theory Fodor-type reflection Greatly Mahlo Partition Relations Slim tree square Ulam matrix Constructible Universe Subadditive Hereditarily Lindelöf space Mandelbrot set sap Sakurai's Bell inequality PFA Poset Commutative projection system Weakly compact cardinal Ineffable cardinal Postprocessing function Erdos-Hajnal graphs Hindman's Theorem full tree coloring number Martin's Axiom Subnormal ideal Closed coloring strongly bounded groups Subtle cardinal Intersection model Chromatic number Fat stationary set Strong coloring Subtle tree property SNR Generalized Clubs O-space Singular cardinals combinatorics Chang's conjecture Microscopic Approach HOD Hedetniemi's conjecture Diamond-sharp Strongly Luzin set Analytic sets Respecting tree Cohen real ZFC construction 54G20 Uniformly homogeneous stationary hitting Local Club Condensation. Commutative cancellative semigroups Small forcing Forcing Axioms Precaliber Diamond S-Space Large Cardinals Lipschitz reduction Prikry-type forcing
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
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
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
Partitioning the club guessing
In a recent paper, I am making use of the following fact. Theorem (Shelah, 1997). Suppose that
Syndetic colorings with applications to S and L
Notation. Write
The S-space problem, and the cardinal invariant
Recall that an
Posted in Blog, Expository, Open Problems
Tagged Hereditarily Lindelöf space, P-Ideal Dichotomy, PFA(S)[S], S-Space
4 Comments
Shelah’s approachability ideal (part 2)
In a previous post, we defined Shelah’s approachability ideal
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
