### 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
- Genearlizations 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

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

# Tag Archives: xbox

## 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 Preprints, Souslin Hypothesis
Tagged 03E05, 03E35, Souslin Tree, square, Weakly compact cardinal, xbox
15 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

## Square with built-in diamond-plus

Joint work with Ralf Schindler. Abstract. We formulate combinatorial principles that combine the square principle with various strong forms of diamond, and prove that the strongest amongst them holds in $L$ for every infinite cardinal. As an application, we prove that … Continue reading

Posted in Publications, Squares and Diamonds
Tagged 03E05, 03E45, Almost Souslin, diamond star, Kurepa Hypothesis, Minimal Walks, square, xbox
1 Comment

## Square principles

Since the birth of Jensen’s original Square principle, many variations of the principle were introduced and intensively studied. Asaf Karagila suggested me today to put some order into all of these principles. Here is a trial. Definition. A square principle … Continue reading