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### Recent blog posts

- 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
- Syndetic colorings with applications to S and L October 26, 2013
- Open coloring and the cardinal invariant $\mathfrak b$ October 8, 2013

### Keywords

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

# 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