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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

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

# 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