### Archives

### Recent blog posts

- A strong form of König’s lemma October 21, 2017
- 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

### Keywords

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

# 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