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### Recent blog posts

- More notions of forcing 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

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

# Tag Archives: Club Guessing

## Partitioning the club guessing

In a recent paper, I am making use of the following fact. Theorem (Shelah, 1997). Suppose that $\kappa$ is an accessible cardinal (i.e., there exists a cardinal $\theta<\kappa$ such that $2^\theta\ge\kappa)$. Then there exists a sequence $\langle g_\delta:C_\delta\rightarrow\omega\mid \delta\in E^{\kappa^+}_\kappa\rangle$ … Continue reading

## Shelah’s approachability ideal (part 2)

In a previous post, we defined Shelah’s approachability ideal $I[\lambda]$. We remind the reader that a subset $S\subseteq\lambda$ is in $I[\lambda]$ iff there exists a collection $\{ \mathcal D_\alpha\mid\alpha<\lambda\}\subseteq\mathcal [\mathcal P(\lambda)]^{<\lambda}$ such that for club many $\delta\in S$, the union … Continue reading

Posted in Blog, Expository, Open Problems
Tagged approachability ideal, Club Guessing
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## Shelah’s approachability ideal (part 1)

Given an infinite cardinal $\lambda$, Shelah defines an ideal $I[\lambda]$ as follows. Definition (Shelah, implicit in here). A set $S$ is in $I[\lambda]$ iff $S\subseteq\lambda$ and there exists a collection $\{ \mathcal D_\alpha\mid\alpha<\lambda\}\subseteq\mathcal [\mathcal P(\lambda)]^{<\lambda}$, and some club $E\subseteq\lambda$, so … Continue reading

## An inconsistent form of club guessing

In this post, we shall present an answer (due to P. Larson) to a question by A. Primavesi concerning a certain strong form of club guessing. We commence with recalling Shelah’s concept of club guessing. Concept (Shelah). Given a regular … Continue reading

## Jensen’s diamond principle and its relatives

This is chapter 6 in the book Set Theory and Its Applications (ISBN: 0821848127). Abstract: We survey some recent results on the validity of Jensen’s diamond principle at successor cardinals. We also discuss weakening of this principle such as club … 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

## Transforming rectangles into squares, with applications to strong colorings

Abstract: It is proved that every singular cardinal $\lambda$ admits a function $\textbf{rts}:[\lambda^+]^2\rightarrow[\lambda^+]^2$ that transforms rectangles into squares. That is, whenever $A,B$ are cofinal subsets of $\lambda^+$, we have $\textbf{rts}[A\circledast B]\supseteq C\circledast C$, for some cofinal subset $C\subseteq\lambda^+$. As a … Continue reading

## The Ostaszewski square, and homogeneous Souslin trees

Abstract: Assume GCH and let $\lambda$ denote an uncountable cardinal. We prove that if $\square_\lambda$ holds, then this may be witnessed by a coherent sequence $\left\langle C_\alpha \mid \alpha<\lambda^+\right\rangle$ with the following remarkable guessing property: For every sequence $\langle A_i\mid i<\lambda\rangle$ … Continue reading

Posted in Publications, Souslin Hypothesis, Squares and Diamonds
Tagged 03E05, 03E35, Club Guessing, Fat stationary set, Ostaszewski square, Souslin Tree
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