### Archives

### Recent blog posts

- A strong form of König’s lemma October 21, 2017
- Prikry forcing may add a Souslin tree June 12, 2016
- The reflection principle $R_2$ May 20, 2016
- Prolific Souslin trees March 17, 2016
- Generalizations 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

### Keywords

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

# 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

Posted in Compactness
Tagged approachability ideal, Aronszajn tree, stationary reflection, Weakly compact cardinal
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## 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 Publications, Souslin Hypothesis
Tagged 03E05, 03E35, 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