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### Recent blog posts

- 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
- Syndetic colorings with applications to S and L October 26, 2013
- Open coloring and the cardinal invariant $\mathfrak b$ October 8, 2013

### Keywords

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

# 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
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## 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