### Archives

### 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

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

# 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
2 Comments

## 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