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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

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

# 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