### 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
- Genearlizations 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

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

# Category Archives: Preprints

## The eightfold way

Joint work with James Cummings, Sy-David Friedman, Menachem Magidor, and Dima Sinapova. Abstract. Three central combinatorial properties in set theory are the tree property, the approachability property and stationary reflection. We prove the mutual independence of these properties by showing … Continue reading

## Reflection on the coloring and chromatic numbers

Joint work with Chris Lambie-Hanson. Abstract. We prove that reflection of the coloring number of a graph is consistent with non-reflection of the chromatic number. Moreover, it is proved that incompactness for the chromatic number of graphs (with arbitrarily large … Continue reading

## 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

## Ordinal definable subsets of singular cardinals

Joint work with James Cummings, Sy-David Friedman, Menachem Magidor, and Dima Sinapova. Abstract. A remarkable result by Shelah states that if $\kappa$ is a singular strong limit cardinal of uncountable cofinality then there is a subset $x$ of $\kappa$ such … Continue reading

Posted in Preprints, Singular Cardinals Combinatorics
Tagged HOD, Singular cardinals combinatorics
1 Comment

## 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 Preprints, Souslin Hypothesis
Tagged 03E05, 03E35, Souslin Tree, square, Weakly compact cardinal, xbox
15 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