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- 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
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- Walk on countable ordinals: the characteristics December 1, 2013
- Polychromatic colorings November 26, 2013
- Universal binary sequences November 14, 2013

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

# Tag Archives: approachability ideal

## Same Graph, Different Universe

Abstract. May the same graph admit two different chromatic numbers in two different universes? how about infinitely many different values? and can this be achieved without changing the cardinals structure? In this paper, it is proved that in Godel’s constructible … Continue reading

Posted in Infinite Graphs, Publications
Tagged 03E35, 05C15, 05C63, approachability ideal, Chromatic number, Constructible Universe, Forcing, Ostaszewski square
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## 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

## 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