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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

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

# 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