### 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

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

# 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 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