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

# 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