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