### Archives

### Recent blog posts

- Prikry forcing may add a Souslin tree June 12, 2016
- The reflection principle $R_2$ May 20, 2016
- Prolific Souslin trees March 17, 2016
- Genearlizations of Martin’s Axiom and the well-met condition January 11, 2015
- Many diamonds from just one January 6, 2015
- Happy new jewish year! September 24, 2014
- Square principles April 19, 2014
- Partitioning the club guessing January 22, 2014

### Keywords

Cardinal function weak diamond coloring number Knaster PFA(S)[S] Souslin Tree Selective Ultrafilter Generalized Clubs Stevo Todorcevic polarized partition relation PFA Fodor-type reflection Successor of Singular Cardinal Prevalent singular cardinals 05A17 11P99 diamond star Almost countably chromatic b-scale tensor product graph Shelah's Strong Hypothesis Small forcing ccc Singular cardinals combinatorics weak square Erdos-Hajnal graphs Club Guessing Rado's conjecture Diamond Fat stationary set Hedetniemi's conjecture 20M14 Commutative cancellative semigroups incompactness Chromatic number Rock n' Roll Dushnik-Miller Weakly compact cardinal Non-saturation Coherent tree Reduced Power Martin's Axiom free Boolean algebra L-space approachability ideal Universal Sequences Almost-disjoint famiy Foundations P-Ideal Dichotomy Successor of Regular Cardinal xbox Cohen real 05D10 Large Cardinals reflection principles sap Antichain Ostaszewski square OCA Rainbow sets stationary hitting Partition Relations Sakurai's Bell inequality Fast club Hereditarily Lindelöf space Jonsson cardinal Prikry-type forcing Axiom R Forcing Axioms Erdos Cardinal projective Boolean algebra Singular Density Constructible Universe Kurepa Hypothesis middle diamond Hindman's Theorem Almost Souslin Mandelbrot set Square-Brackets Partition Relations Chang's conjecture Minimal Walks Microscopic Approach Singular Cofinality Ascent Path Uniformization Poset S-Space square Singular coﬁnality very good scale Whitehead Problem Aronszajn tree Absoluteness stationary reflection Slim tree Parameterized proxy principle HOD Cardinal Invariants Forcing

# Tag Archives: Weakly compact cardinal

## 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

## 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 Preprints, Souslin Hypothesis
Tagged 03E05, 03E35, 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
2 Comments