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