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

# 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