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### Keywords

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

# 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