### 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

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

# 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 Publications, 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
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