### 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
- Genearlizations 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

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

# 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 Preprints, Souslin Hypothesis
Tagged 03E05, 03E35, Souslin Tree, square, Weakly compact cardinal, xbox
15 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 Preprints, Souslin Hypothesis
Tagged 03E05, 03E35, 03E65, 05C05, Coherent tree, Diamond, Microscopic Approach, Parameterized proxy principle, Slim tree, Souslin Tree, square, xbox
3 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