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### Recent blog posts

- 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
- Gabriel Belachsan (14/5/1976 – 20/8/2013) August 20, 2013

### Keywords

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

# Tag Archives: stationary hitting

## 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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## 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