Archives
Keywords
Singular cofinality Nonspecial tree b-scale Amenable C-sequence Diamond Partition Relations Uniformly homogeneous Uniformly coherent Axiom R Intersection model Analytic sets P-Ideal Dichotomy very good scale Dushnik-Miller Partition relations for trees Subnormal ideal free Souslin tree nonmeager set Subtle cardinal Monotonically far incompactness middle diamond weak Kurepa tree Respecting tree Square-Brackets Partition Relations Forcing Absoluteness Subadditive reflection principles Constructible Universe weak diamond Rock n' Roll Martin's Axiom Reduced Power xbox Generalized descriptive set theory Erdos-Hajnal graphs Hereditarily Lindelöf space Small forcing Cohen real L-space Almost-disjoint family S-Space Open Access Lipschitz reduction Cardinal function Kurepa Hypothesis Universal Sequences transformations stick sap Strongly Luzin set stationary hitting Postprocessing function stationary reflection Poset Rainbow sets ZFC construction Ascent Path Ulam matrix Greatly Mahlo GMA Antichain O-space diamond star Shelah's Strong Hypothesis Diamond for trees super-Souslin tree Closed coloring AIM forcing Hedetniemi's conjecture higher Baire space Uniformization Prikry-type forcing Singular cardinals combinatorics Chromatic number coloring number PFA(S)[S] Souslin Tree Successor of Singular Cardinal Ostaszewski square Strong coloring Distributive tree Slim tree projective Boolean algebra Almost countably chromatic Cardinal Invariants Diamond-sharp Knaster Well-behaved magma Countryman line Forcing with side conditions Foundations Local Club Condensation. Chang's conjecture Whitehead Problem positive partition relation 54G20 Filter reflection indecomposable filter perfectly normal Reflecting stationary set Selective Ultrafilter Knaster and friends SNR unbounded function tensor product graph Dowker space Prevalent singular cardinals square Singular Density Vanishing levels Microscopic Approach Commutative cancellative semigroups Precaliber PFA Non-saturation approachability ideal Strongly compact cardinal Aronszajn tree Fast club Erdos Cardinal polarized partition relation Fodor-type reflection regressive Souslin tree OCA Sigma-Prikry Commutative projection system Parameterized proxy principle Mandelbrot set countably metacompact HOD Ineffable cardinal Coherent tree Rado's conjecture Interval topology on trees Luzin set Large Cardinals Iterated forcing weak square Ramsey theory over partitions club_AD free Boolean algebra Club Guessing Jonsson cardinal Weakly compact cardinal full tree square principles Entangled linear order strongly bounded groups Successor of Regular Cardinal Hindman's Theorem specializable Souslin tree Fat stationary set Sierpinski's onto mapping principle Generalized Clubs Was Ulam right? ccc Almost Souslin Subtle tree property Ascending path C-sequence Forcing Axioms Minimal Walks Sakurai's Bell inequality
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
1 Comment
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
Comments Off on The reflection principle $R_2$
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
2 Comments
