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### Recent blog posts

- Genearlizations 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
- Walk on countable ordinals: the characteristics December 1, 2013
- Polychromatic colorings November 26, 2013
- Universal binary sequences November 14, 2013

### Keywords

stationary reflection Aronszajn tree Prikry-type forcing Erdos Cardinal Non-saturation Forcing Singular cardinals combinatorics diamond star P-Ideal Dichotomy b-scale Cardinal Invariants Club Guessing Large Cardinals Erdos-Hajnal graphs reflection principles Successor of Regular Cardinal Hedetniemi's conjecture square Shelah's Strong Hypothesis Dushnik-Miller Cohen real Hereditarily Lindelöf space Rado's conjecture Partition Relations Sakurai's Bell inequality ccc PFA(S)[S] projective Boolean algebra Small forcing Knaster Prevalent singular cardinals incompactness Almost-disjoint famiy Ostaszewski square Uniformization approachability ideal sap Diamond Absoluteness weak square Rock n' Roll Whitehead Problem Martin's Axiom PFA Singular Cofinality OCA Universal Sequences Forcing Axioms Weakly compact cardinal Mandelbrot set Cardinal function Antichain L-space Foundations middle diamond Rainbow sets Souslin Tree Constructible Universe Square-Brackets Partition Relations Generalized Clubs tensor product graph Axiom R free Boolean algebra weak diamond very good scale stationary hitting Singular Density Minimal Walks Almost countably chromatic Poset Successor of Singular Cardinal Chromatic number polarized partition relation S-Space Kurepa Hypothesis

# Tag Archives: sap

## 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

Posted in Open Problems, Publications
Tagged 03E05, 03E35, 03E50, approachability ideal, Club Guessing, Diamond, diamond star, Non-saturation, sap, Souslin Tree, square, stationary hitting, Uniformization, Whitehead Problem
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## 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