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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

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

# 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