### 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

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

## 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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