### 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

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

# 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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