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

# 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