### 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

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

# 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