### 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

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

# 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