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Was Ulam right? Nonspecial tree Prevalent singular cardinals transformations Fast club Ascent Path indecomposable filter S-Space Chromatic number Axiom R Singular cardinals combinatorics Sierpinski's onto mapping principle Amenable C-sequence Analytic sets b-scale GMA Singular cofinality Weakly compact cardinal square L-space Distributive tree Club Guessing Non-saturation PFA stick Ineffable cardinal Erdos-Hajnal graphs Minimal Walks incompactness SNR Universal Sequences Subadditive stationary reflection Partition Relations C-sequence Lipschitz reduction Intersection model Postprocessing function Filter reflection Kurepa Hypothesis Uniformly coherent P-Ideal Dichotomy ccc Ulam matrix Precaliber free Souslin tree Poset Antichain Large Cardinals Countryman line coloring number strongly bounded groups Fat stationary set Microscopic Approach projective Boolean algebra approachability ideal Reduced Power Almost-disjoint family Parameterized proxy principle Jonsson cardinal Interval topology on trees higher Baire space free Boolean algebra Diamond for trees square principles Sigma-Prikry Rado's conjecture stationary hitting Rock n' Roll Ascending path Hedetniemi's conjecture Vanishing levels Entangled linear order diamond star Sakurai's Bell inequality Iterated forcing unbounded function Monotonically far Singular Density Respecting tree club_AD countably metacompact Forcing with side conditions Well-behaved magma weak diamond Ramsey theory over partitions Greatly Mahlo Knaster O-space OCA Aronszajn tree Dowker space Small forcing Forcing Square-Brackets Partition Relations weak square Martin's Axiom reflection principles Chang's conjecture Generalized descriptive set theory Coherent tree Foundations polarized partition relation Successor of Regular Cardinal Uniformization Selective Ultrafilter Commutative projection system Cardinal Invariants Whitehead Problem ZFC construction positive partition relation super-Souslin tree Erdos Cardinal Subtle tree property tensor product graph perfectly normal Rainbow sets Subnormal ideal Forcing Axioms Constructible Universe middle diamond Cohen real Slim tree Fodor-type reflection Open Access Dushnik-Miller Commutative cancellative semigroups xbox Almost Souslin Mandelbrot set Reflecting stationary set Hereditarily Lindelöf space Ostaszewski square Hindman's Theorem Almost countably chromatic nonmeager set weak Kurepa tree Generalized Clubs AIM forcing Cardinal function Prikry-type forcing PFA(S)[S] Strongly Luzin set Knaster and friends Souslin Tree Strong coloring Diamond-sharp Uniformly homogeneous Diamond Local Club Condensation. Luzin set HOD very good scale full tree Absoluteness Successor of Singular Cardinal Subtle cardinal Strongly compact cardinal regressive Souslin tree sap 54G20 Partition relations for trees specializable Souslin tree Closed coloring Shelah's Strong Hypothesis
Category Archives: Compactness
A new model for all C-sequences are trivial
Joint work with Zhixing You and Jiachen Yuan. Abstract. We construct a model in which all C-sequences are trivial, yet there exists a $\kappa$-Souslin tree with full vanishing levels. This answers a question from a previous paper, and provides an … Continue reading
Posted in Compactness, Publications
Tagged Ascent Path, C-sequence, Intersection model, Souslin Tree, Subtle tree property, Vanishing levels
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Ketonen’s question and other cardinal sins
Joint work with Zhixing You and Jiachen Yuan. Abstract. Answering a question of Ketonen from the late 1970’s, it is proved that a weakly compact cardinal carrying an indecomposable ultrafilter need not be measurable. The result is obtained by analyzing … Continue reading
Squares, ultrafilters and forcing axioms
Joint work with Chris Lambie-Hanson and Jing Zhang. Abstract. We study the interplay of the three families of combinatorial objects or principles. Specifically, we show the following. Strong forcing axioms, in general incompatible with the existence of indexed squares, can … Continue reading
Sigma-Prikry forcing III: Down to Aleph_omega
Joint work with Alejandro Poveda and Dima Sinapova. Abstract. We prove the consistency of the failure of the singular cardinals hypothesis at $\aleph_\omega$ together with the reflection of all stationary subsets of $\aleph_{\omega+1}$. This shows that two classical results of … Continue reading
Sigma-Prikry forcing II: Iteration Scheme
Joint work with Alejandro Poveda and Dima Sinapova. Abstract. In Part I of this series, we introduced a class of notions of forcing which we call $\Sigma$-Prikry, and showed that many of the known Prikry-type notions of forcing that centers … Continue reading
Knaster and friends II: The C-sequence number
Joint work with Chris Lambie-Hanson. Abstract. Motivated by a characterization of weakly compact cardinals due to Todorcevic, we introduce a new cardinal characteristic, the C-sequence number, which can be seen as a measure of the compactness of a regular uncountable … Continue reading
Sigma-Prikry forcing I: The Axioms
Joint work with Alejandro Poveda and Dima Sinapova. Abstract. We introduce a class of notions of forcing which we call $\Sigma$-Prikry, and show that many of the known Prikry-type notions of forcing that centers around singular cardinals of countable cofinality … Continue reading
The eightfold way
Joint work with James Cummings, Sy-David Friedman, Menachem Magidor, and Dima Sinapova. Abstract. Three central combinatorial properties in set theory are the tree property, the approachability property and stationary reflection. We prove the mutual independence of these properties by showing … Continue reading
Reflection on the coloring and chromatic numbers
Joint work with Chris Lambie-Hanson. Abstract. We prove that reflection of the coloring number of graphs is consistent with non-reflection of the chromatic number. Moreover, it is proved that incompactness for the chromatic number of graphs (with arbitrarily large gaps) … Continue reading
Posted in Compactness, Infinite Graphs, Publications
Tagged 03E35, 05C15, 05C63, Chang's conjecture, Chromatic number, coloring number, Fodor-type reflection, incompactness, Iterated forcing, Parameterized proxy principle, Postprocessing function, Rado's conjecture, square, stationary reflection
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