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