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- Prikry forcing may add a Souslin tree June 12, 2016
- The reflection principle $R_2$ May 20, 2016
- Prolific Souslin trees March 17, 2016
- 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
- Square principles April 19, 2014
- Partitioning the club guessing January 22, 2014

### Keywords

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

# 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