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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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### Keywords

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

# 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