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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
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### Keywords

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

# 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? We answer these questions in the affirmative. In this paper, … Continue reading

Posted in Preprints
Tagged 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

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