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