### 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

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

# 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