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### Keywords

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

# 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