### Archives

### Recent blog posts

- 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
- Gabriel Belachsan (14/5/1976 – 20/8/2013) August 20, 2013

### Keywords

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

# 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