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