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