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