### 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

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

# 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