### 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

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

# 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