### Archives

### 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

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

# 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

## 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