### 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

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

# 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 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

## 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