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### 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

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

# 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
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## 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
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## 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