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

# 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

Posted in Compactness
Tagged approachability ideal, Aronszajn tree, stationary reflection, Weakly compact cardinal
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## 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