### 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
- 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
- Partitioning the club guessing January 22, 2014

### Keywords

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

# 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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