### 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

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

# 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