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- Prikry forcing may add a Souslin tree June 12, 2016
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weak diamond Prikry-type forcing Singular coﬁnality Jonsson cardinal Mandelbrot set Poset reflection principles sap Hindman's Theorem xbox Cohen real Partition Relations Non-saturation b-scale Martin's Axiom Ostaszewski square Erdos Cardinal middle diamond PFA Diamond stationary reflection Universal Sequences Chang's conjecture polarized partition relation Slim tree Commutative cancellative semigroups approachability ideal Ascent Path stationary hitting Forcing Axioms Whitehead Problem Successor of Regular Cardinal Singular Density 05D10 Foundations Absoluteness Coherent tree Selective Ultrafilter Parameterized proxy principle Constructible Universe 05A17 square Sakurai's Bell inequality Dushnik-Miller Hereditarily Lindelöf space Aronszajn tree Almost-disjoint famiy Forcing Rock n' Roll Large Cardinals Small forcing Minimal Walks Weakly compact cardinal projective Boolean algebra PFA(S)[S] Successor of Singular Cardinal ccc Fodor-type reflection Chromatic number weak square Club Guessing Antichain tensor product graph Rado's conjecture 20M14 Singular Cofinality S-Space Almost Souslin Souslin Tree Almost countably chromatic Generalized Clubs Microscopic Approach Stevo Todorcevic diamond star Fat stationary set Shelah's Strong Hypothesis Cardinal Invariants Rainbow sets Reduced Power Erdos-Hajnal graphs coloring number Hedetniemi's conjecture HOD Singular cardinals combinatorics OCA very good scale L-space Fast club Square-Brackets Partition Relations Knaster P-Ideal Dichotomy Cardinal function free Boolean algebra Axiom R 11P99 Uniformization Prevalent singular cardinals incompactness Kurepa Hypothesis

# Tag Archives: approachability ideal

## The eightfold way

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

## 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