### 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

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

# 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
2 Comments

## 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