### 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

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

# 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