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