### 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

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

# Category Archives: Souslin Hypothesis

## More notions of forcing add a Souslin tree

Joint work with Ari Meir Brodsky. Abstract. An $\aleph_1$-Souslin tree is a complicated combinatorial object whose existence cannot be decided on the grounds of ZFC alone. But 15 years after Tennenbaum and independently Jech devised notions of forcing for introducing … Continue reading

## 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
16 Comments

## A Microscopic approach to Souslin-tree constructions. Part I

Joint work with Ari Meir Brodsky. Abstract. We propose a parameterized proxy principle from which $\kappa$-Souslin trees with various additional features can be constructed, regardless of the identity of $\kappa$. We then introduce the microscopic approach, which is a simple … Continue reading

Posted in Publications, Souslin Hypothesis
Tagged 03E05, 03E35, 03E65, 05C05, Coherent tree, Diamond, Microscopic Approach, Parameterized proxy principle, Slim tree, Souslin Tree, square, xbox
4 Comments

## Reduced powers of Souslin trees

Joint work with Ari Meir Brodsky. Abstract. We study the relationship between a $\kappa$-Souslin tree $T$ and its reduced powers $T^\theta/\mathcal U$. Previous works addressed this problem from the viewpoint of a single power $\theta$, whereas here, tools are developed … Continue reading

## On guessing generalized clubs at the successors of regulars

Abstract: Konig, Larson and Yoshinobu initiated the study of principles for guessing generalized clubs, and introduced a construction of an higher Souslin tree from the strong guessing principle. Complementary to the author’s work on the validity of diamond and non-saturation … Continue reading

## The Ostaszewski square, and homogeneous Souslin trees

Abstract: Assume GCH and let $\lambda$ denote an uncountable cardinal. We prove that if $\square_\lambda$ holds, then this may be witnessed by a coherent sequence $\left\langle C_\alpha \mid \alpha<\lambda^+\right\rangle$ with the following remarkable guessing property: For every sequence $\langle A_i\mid i<\lambda\rangle$ … Continue reading

Posted in Publications, Souslin Hypothesis, Squares and Diamonds
Tagged 03E05, 03E35, Club Guessing, Fat stationary set, Ostaszewski square, Souslin Tree
5 Comments