### 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

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

# 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