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### Recent blog posts

- Partitioning the club guessing January 22, 2014
- Walk on countable ordinals: the characteristics December 1, 2013
- Polychromatic colorings November 26, 2013
- Universal binary sequences November 14, 2013
- Syndetic colorings with applications to S and L October 26, 2013
- Open coloring and the cardinal invariant $\mathfrak b$ October 8, 2013
- Gabriel Belachsan (14/5/1976 – 20/8/2013) August 20, 2013
- PFA and the tree property at $\aleph_2$ June 9, 2013

### Keywords

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

- Luzin sets and generalizations
- Nonuniversal colorings in ZFC
- Large Sets
- Infinite-dimensional Jonsson algebras
- Strong colorings without nontrivial polychromatic sets
- Infinite-dimensional polychromatic colorings
- Polychromatic colorings of the first uncountable cardinal
- From colorings to topology
- From topology to colorings
- Anti-Ramsey colorings of the rational numbers, part 2

# 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

Posted in Open Problems, Publications
Tagged 03E05, 03E35, 03E50, approachability ideal, Club Guessing, Diamond, diamond star, Non-saturation, sap, Souslin Tree, square, stationary hitting, Uniformization, Whitehead Problem
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## 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
Tagged 03E05, 03E35, Club Guessing, Ostaszewski square, Souslin Tree
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