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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
- Generalizations 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
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- Partitioning the club guessing January 22, 2014

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

# 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