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- Genearlizations of Martin’s Axiom and the well-met condition January 11, 2015
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- Happy new jewish year! September 24, 2014
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### Keywords

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