### 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

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