### Archives

### Recent blog posts

- 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
- Walk on countable ordinals: the characteristics December 1, 2013
- Polychromatic colorings November 26, 2013

### Keywords

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

# 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