### 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

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

# 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