### 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

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

# Tag Archives: 03E65

## A remark on Schimmerling’s question

Joint work with Ari Meir Brodsky. Abstract. Schimmerling asked whether $\square^*_\lambda$ together with GCH entails the existence of a $\lambda^+$-Souslin tree, for a singular cardinal $\lambda$. Here, we provide an affirmative answer under the additional assumption that there exists a … Continue reading

## 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

## Reduced powers of Souslin trees

Joint work with Ari Meir Brodsky. Abstract. We study the relationship between a $\kappa$-Souslin tree $T$ and its reduced powers $T^\theta/\mathcal U$. Previous works addressed this problem from the viewpoint of a single power $\theta$, whereas here, tools are developed … Continue reading

## A topological reflection principle equivalent to Shelah’s strong hypothesis

Abstract: We notice that Shelah’s Strong Hypothesis (SSH) is equivalent to the following reflection principle: Suppose $\mathbb X$ is an (infinite) first-countable space whose density is a regular cardinal, $\kappa$. If every separable subspace of $\mathbb X$ is of cardinality at most … Continue reading

Posted in Compactness, Publications, Topology
Tagged 03E04, 03E65, 54G15, Shelah's Strong Hypothesis
Leave a comment

## Openly generated Boolean algebras and the Fodor-type reflection principle

Joint work with Sakaé Fuchino. Abstract: We prove that the Fodor-type Reflection Principle (FRP) is equivalent to the assertion that any Boolean algebra is openly generated if and only if it is $\aleph _2$-projective. Previously it was known that this … Continue reading

## On guessing generalized clubs at the successors of regulars

Abstract: Konig, Larson and Yoshinobu initiated the study of principles for guessing generalized clubs, and introduced a construction of an higher Souslin tree from the strong guessing principle. Complementary to the author’s work on the validity of diamond and non-saturation … Continue reading

## On the consistency strength of the Milner-Sauer conjecture

Abstract: In their paper from 1981, Milner and Sauer conjectured that for any poset $\mathbb P$, if $\text{cf}(\mathbb P)$ is a singular cardinal $\lambda$, then $\mathbb P$ must contain an antichain of size $\text{cf}(\lambda)$. The conjecture is consistent and known … Continue reading