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Whitehead Problem Almost-disjoint famiy OCA polarized partition relation Chromatic number Shelah's Strong Hypothesis Mandelbrot set Fast club Selective Ultrafilter middle diamond Fat stationary set Knaster Slim tree Erdos Cardinal b-scale Microscopic Approach Cohen real Non-saturation stationary hitting Parameterized proxy principle Erdos-Hajnal graphs Successor of Regular Cardinal sap reflection principles Partition Relations HOD Absoluteness Rainbow sets Prevalent singular cardinals Large Cardinals Chang's conjecture PFA Small forcing Square-Brackets Partition Relations 20M14 Poset Souslin Tree Almost Souslin Kurepa Hypothesis Almost countably chromatic Singular cardinals combinatorics diamond star 11P99 Generalized Clubs Reduced Power Ascent Path Jonsson cardinal Aronszajn tree incompactness Martin's Axiom 05D10 Rock n' Roll square projective Boolean algebra Cardinal function Uniformization Stevo Todorcevic Successor of Singular Cardinal 05A17 Commutative cancellative semigroups Constructible Universe L-space tensor product graph Prikry-type forcing Diamond Weakly compact cardinal S-Space weak square Hereditarily Lindelöf space Foundations Singular Density Forcing Axioms ccc approachability ideal weak diamond Singular Cofinality Antichain stationary reflection Sakurai's Bell inequality PFA(S)[S] Singular coﬁnality Forcing Ostaszewski square very good scale Cardinal Invariants Dushnik-Miller Club Guessing Universal Sequences Axiom R Rado's conjecture Coherent tree free Boolean algebra Hedetniemi's conjecture Minimal Walks Hindman's Theorem P-Ideal Dichotomy

# 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