### 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

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

# 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