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### 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

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

# 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