Tag Archives: approachability ideal

The eightfold way

Posted on December 23, 2016 by Assaf Rinot in categories: Compactness.

Joint work with James Cummings, Sy-David Friedman, Menachem Magidor, and Dima Sinapova. Abstract. Three central combinatorial properties in set theory are the tree property, the approachability property and stationary reflection. We prove the mutual independence of these properties by showing … Continue reading →

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Same Graph, Different Universe

Posted on December 13, 2014 by Assaf Rinot in categories: Infinite Graphs, Publications.

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 →

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Shelah’s approachability ideal (part 2)

Posted on October 4, 2012 by Assaf Rinot in categories: Blog, Expository, Open Problems.

In a previous post, we defined Shelah’s approachability ideal I[λ]. We remind the reader that a subset S⊆λ is in I[λ] iff there exists a collection {Dα∣α<λ}⊆[P(λ)]<λ such that for club many δ∈S, the union … Continue reading →

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Shelah’s approachability ideal (part 1)

Posted on April 24, 2012 by Assaf Rinot in categories: Blog, Expository.

Given an infinite cardinal λ, Shelah defines an ideal I[λ] as follows. Definition (Shelah, implicit in here). A set S is in I[λ] iff S⊆λ and there exists a collection {Dα∣α<λ}⊆[P(λ)]<λ, and some club E⊆λ, so … Continue reading →

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Jensen’s diamond principle and its relatives

Posted on October 6, 2011 by Assaf Rinot in categories: Open Problems, Publications, Squares and Diamonds.

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 →

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The failure of diamond on a reflecting stationary set

Posted on September 27, 2011 by Assaf Rinot in categories: Publications, Squares and Diamonds.

Joint work with Moti Gitik. Abstract: It is shown that the failure of ♢S, for a subset S⊆ℵω+1 that reflects stationarily often, is consistent with GCH and APℵω, relatively to the existence of a supercompact cardinal. This should be comapred with … Continue reading →

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A relative of the approachability ideal, diamond and non-saturation

Posted on September 27, 2011 by Assaf Rinot in categories: Publications, Squares and Diamonds.

Abstract: Let λ denote a singular cardinal. Zeman, improving a previous result of Shelah, proved that ◻λ∗ together with 2λ=λ+ implies ♢S for every S⊆λ+ that reflects stationarily often. In this paper, for a subset S⊂λ+, a normal subideal of … Continue reading →

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