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

# 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