### 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

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

# Category Archives: Souslin Hypothesis

## A forcing axiom deciding the generalized Souslin Hypothesis

Joint work with Chris Lambie-Hanson. Abstract. We derive a forcing axiom from the conjunction of square and diamond, and present a few applications, primary among them being the existence of super-Souslin trees. It follows that for every uncountable cardinal $\lambda$, … Continue reading

Posted in Preprints, Souslin Hypothesis
Tagged 03E05, 03E35, 03E57, Diamond, Forcing Axioms, Souslin Tree, square, super-Souslin tree
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## 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
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## 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