### 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

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

# 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