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Slim tree Diamond-sharp Distributive tree Souslin Tree Uniformly homogeneous Monotonically far Strongly compact cardinal indecomposable filter Dushnik-Miller Was Ulam right? PFA(S)[S] Entangled linear order Greatly Mahlo P-Ideal Dichotomy Respecting tree Prikry-type forcing diamond star Poset unbounded function Intersection model Subtle tree property Coherent tree Ulam matrix Ascent Path very good scale Forcing Axioms Subnormal ideal Shelah's Strong Hypothesis Uniformization club_AD OCA Foundations Erdos Cardinal Large Cardinals Axiom R stationary hitting Vanishing levels Subadditive super-Souslin tree reflection principles Knaster Rainbow sets approachability ideal Hereditarily Lindelöf space xbox higher Baire space HOD Diamond for trees Chang's conjecture Ramsey theory over partitions Strong coloring AIM forcing Almost countably chromatic Aronszajn tree C-sequence GMA Iterated forcing Forcing with side conditions Fodor-type reflection b-scale Subtle cardinal ZFC construction Parameterized proxy principle Postprocessing function Rock n' Roll Almost-disjoint family weak diamond 54G20 Uniformly coherent full tree incompactness square principles Generalized descriptive set theory Singular cofinality free Boolean algebra transformations Open Access Microscopic Approach PFA Hindman's Theorem L-space Partition relations for trees Reflecting stationary set weak square Weakly compact cardinal perfectly normal Ineffable cardinal Interval topology on trees specializable Souslin tree Sigma-Prikry strongly bounded groups Prevalent singular cardinals projective Boolean algebra sap Ascending path Filter reflection Ostaszewski square Cohen real Successor of Singular Cardinal Selective Ultrafilter Forcing countably metacompact SNR Rado's conjecture Amenable C-sequence Strongly Luzin set Diamond Universal Sequences Successor of Regular Cardinal Jonsson cardinal Kurepa Hypothesis Square-Brackets Partition Relations stick Commutative cancellative semigroups Precaliber Cardinal function Reduced Power Constructible Universe Fast club Dowker space Antichain Hedetniemi's conjecture S-Space Lipschitz reduction Absoluteness weak Kurepa tree square Nonspecial tree polarized partition relation Club Guessing tensor product graph Partition Relations Martin's Axiom Erdos-Hajnal graphs Singular Density Commutative projection system Whitehead Problem Local Club Condensation. Chromatic number coloring number middle diamond positive partition relation regressive Souslin tree free Souslin tree O-space Mandelbrot set Small forcing Analytic sets Singular cardinals combinatorics Non-saturation Generalized Clubs Almost Souslin Luzin set Minimal Walks Sierpinski's onto mapping principle Sakurai's Bell inequality ccc Countryman line Fat stationary set nonmeager set Well-behaved magma Closed coloring Cardinal Invariants stationary reflection Knaster and friends
Tag Archives: Weakly compact cardinal
Was Ulam right? I: Basic theory and subnormal ideals
Joint work with Tanmay Inamdar. Abstract. We introduce various coloring principles which generalize the so-called onto mapping principle of Sierpinski to larger cardinals and general ideals. We prove that these principles capture the notion of an Ulam matrix and allow … Continue reading
Fake Reflection
Joint work with Gabriel Fernandes and Miguel Moreno. Abstract. We introduce a generalization of stationary set reflection which we call filter reflection, and show it is compatible with the axiom of constructibility as well as with strong forcing axioms. We … 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
Strong failures of higher analogs of Hindman’s Theorem
Joint work with David J. Fernández Bretón. Abstract. We show that various analogs of Hindman’s Theorem fail in a strong sense when one attempts to obtain uncountable monochromatic sets: Theorem 1. There exists a colouring $c:\mathbb R\rightarrow\mathbb Q$, such that … Continue reading
Posted in Groups, Partition Relations, Publications
Tagged 03E02, 03E35, 03E75, 05A17, 05D10, 11P99, 20M14, Chang's conjecture, Commutative cancellative semigroups, Entangled linear order, Erdos Cardinal, Hindman's Theorem, Jonsson cardinal, Kurepa Hypothesis, Square-Brackets Partition Relations, Weakly compact cardinal, ZFC construction
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The reflection principle $R_2$
A few years ago, in this paper, I introduced the following reflection principle: Definition. $R_2(\theta,\kappa)$ asserts that for every function $f:E^\theta_{<\kappa}\rightarrow\kappa$, there exists some $j<\kappa$ for which the following set is nonstationary: $$A_j:=\{\delta\in E^\theta_\kappa\mid f^{-1}[j]\cap\delta\text{ is nonstationary}\}.$$ I wrote there … Continue reading
Posted in Blog
Tagged reflection principles, square, stationary reflection, Weakly compact cardinal
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Higher Souslin trees and the GCH, revisited
Abstract. It is proved that for every uncountable cardinal $\lambda$, GCH+$\square(\lambda^+)$ entails the existence of a $\text{cf}(\lambda)$-complete $\lambda^+$-Souslin tree. In particular, if GCH holds and there are no $\aleph_2$-Souslin trees, then $\aleph_2$ is weakly compact in Godel’s constructible universe, improving … Continue reading
Posted in Publications, Souslin Hypothesis
Tagged 03E05, 03E35, Open Access, regressive Souslin tree, Souslin Tree, square, Weakly compact cardinal, xbox
16 Comments
Chain conditions of products, and weakly compact cardinals
Abstract. The history of productivity of the $\kappa$-chain condition in partial orders, topological spaces, or Boolean algebras is surveyed, and its connection to the set-theoretic notion of a weakly compact cardinal is highlighted. Then, it is proved that for every … Continue reading
Posted in Partition Relations, Publications
Tagged Aronszajn tree, ccc, Fat stationary set, Minimal Walks, square, Weakly compact cardinal
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