### Archives

### Recent blog posts

- Prikry forcing may add a Souslin tree June 12, 2016
- The reflection principle $R_2$ May 20, 2016
- Prolific Souslin trees March 17, 2016
- 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

### Keywords

Slim tree Erdos-Hajnal graphs Shelah's Strong Hypothesis Prikry-type forcing Souslin Tree 11P99 OCA projective Boolean algebra Successor of Regular Cardinal Martin's Axiom Selective Ultrafilter Forcing Fat stationary set L-space Reduced Power Whitehead Problem Non-saturation stationary hitting Large Cardinals very good scale Ascent Path Successor of Singular Cardinal Sakurai's Bell inequality Singular Density Partition Relations stationary reflection 05D10 Dushnik-Miller Constructible Universe P-Ideal Dichotomy Antichain approachability ideal Stevo Todorcevic Axiom R polarized partition relation reflection principles Almost Souslin Mandelbrot set sap S-Space Aronszajn tree Chromatic number weak diamond Hedetniemi's conjecture tensor product graph Poset Prevalent singular cardinals 05A17 Rado's conjecture weak square Hereditarily Lindelöf space Singular coﬁnality Parameterized proxy principle Ostaszewski square Fast club PFA(S)[S] PFA Almost-disjoint famiy Rainbow sets middle diamond Almost countably chromatic Chang's conjecture Foundations ccc Diamond Cardinal function Cohen real Small forcing HOD Cardinal Invariants diamond star square Commutative cancellative semigroups incompactness free Boolean algebra Singular cardinals combinatorics Hindman's Theorem Absoluteness Knaster 20M14 Universal Sequences Coherent tree Rock n' Roll Kurepa Hypothesis Microscopic Approach Generalized Clubs Forcing Axioms b-scale Singular Cofinality Club Guessing Uniformization Weakly compact cardinal Erdos Cardinal Jonsson cardinal Minimal Walks Square-Brackets Partition Relations

# 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