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### Keywords

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

# Tag Archives: diamond star

## A Kurepa tree from diamond-plus

Recall that $T$ is said to be a $\kappa$-Kurepa tree if $T$ is a tree of height $\kappa$, whose levels $T_\alpha$ has size $\le|\alpha|$ for co-boundedly many $\alpha<\kappa$, and such that the set of branches of $T$ has size $>\kappa$. … Continue reading

## 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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## 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