### 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

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

# 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 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