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- Partitioning the club guessing January 22, 2014
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Shelah's Strong Hypothesis incompactness Forcing Axioms Constructible Universe P-Ideal Dichotomy b-scale Generalized Clubs Prikry-type forcing Diamond Rainbow sets Erdos-Hajnal graphs reflection principles polarized partition relation Club Guessing weak diamond Hereditarily Lindelöf space Rado's conjecture stationary reflection Martin's Axiom tensor product graph Prevalent singular cardinals Absoluteness Dushnik-Miller Axiom R Almost-disjoint famiy Almost countably chromatic Chromatic number Successor of Regular Cardinal very good scale Mandelbrot set Antichain OCA middle diamond Small forcing Universal Sequences Whitehead Problem Cardinal function Non-saturation Square-Brackets Partition Relations Erdos Cardinal Large Cardinals Poset Forcing Uniformization Successor of Singular Cardinal Foundations S-Space square approachability ideal Sakurai's Bell inequality PFA(S)[S] L-space Kurepa Hypothesis Singular cardinals combinatorics Cohen real Singular Cofinality diamond star sap stationary hitting Knaster Rock n' Roll Minimal Walks projective Boolean algebra Ostaszewski square weak square free Boolean algebra Aronszajn tree Singular Density Cardinal Invariants Partition Relations Hedetniemi's conjecture PFA Souslin Tree### Ongoing seminar

- Luzin sets and generalizations
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- 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: approachability ideal

## 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

## 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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## The failure of diamond on a reflecting stationary set

Joint work with Moti Gitik. Abstract: It is shown that the failure of $\diamondsuit_S$, for a subset $S\subseteq\aleph_{\omega+1}$ that reflects stationarily often, is consistent with GCH and $\text{AP}_{\aleph_\omega}$, relatively to the existence of a supercompact cardinal. This should be comapred with … Continue reading

## A relative of the approachability ideal, diamond and non-saturation

Abstract: Let $\lambda$ denote a singular cardinal. Zeman, improving a previous result of Shelah, proved that $\square^*_\lambda$ together with $2^\lambda=\lambda^+$ implies $\diamondsuit_S$ for every $S\subseteq\lambda^+$ that reflects stationarily often. In this paper, for a subset $S\subset\lambda^+$, a normal subideal of … Continue reading