### 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

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

# 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
2 Comments