### Archives

### Recent blog posts

- 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
- Square principles April 19, 2014
- Partitioning the club guessing January 22, 2014

### Keywords

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

# 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
10 Comments

## 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
Leave a comment

## 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