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- More notions of forcing add a Souslin tree June 12, 2016
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- Genearlizations of Martin’s Axiom and the well-met condition January 11, 2015
- Many diamonds from just one January 6, 2015
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HOD Rainbow sets Mandelbrot set Fat stationary set Prikry-type forcing Constructible Universe Singular cardinals combinatorics Forcing diamond star Antichain Erdos Cardinal Fast club Minimal Walks Singular Cofinality Sakurai's Bell inequality weak diamond Ascent Path square polarized partition relation weak square Shelah's Strong Hypothesis Rado's conjecture Diamond stationary hitting Slim tree Generalized Clubs Cardinal Invariants tensor product graph Partition Relations Erdos-Hajnal graphs Forcing Axioms Dushnik-Miller b-scale stationary reflection Non-saturation approachability ideal Almost Souslin free Boolean algebra ccc sap Rock n' Roll Axiom R Martin's Axiom Stevo Todorcevic Large Cardinals L-space Hedetniemi's conjecture Singular Density Souslin Tree Cardinal function Universal Sequences Club Guessing Successor of Singular Cardinal Selective Ultrafilter S-Space Knaster Kurepa Hypothesis Small forcing Microscopic Approach Cohen real Aronszajn tree Successor of Regular Cardinal Absoluteness middle diamond Hereditarily Lindelöf space PFA(S)[S] Uniformization Foundations Coherent tree PFA Chromatic number Weakly compact cardinal Poset Parameterized proxy principle reflection principles Reduced Power projective Boolean algebra Prevalent singular cardinals Whitehead Problem incompactness Singular coﬁnality very good scale Almost-disjoint famiy Ostaszewski square Almost countably chromatic Square-Brackets Partition Relations P-Ideal Dichotomy OCA

# Tag Archives: approachability ideal

## 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