Weak diamond and Galvin's property

Abstract

We prove that the Devlin-Shelah weak diamond implies Galvin's property. On the other hand, Galvin's property is consistent with the negation of the weak diamond, and even with Martin's axiom. We show that the proper forcing axiom implies a relative to the negation of Galvin's property for ${\aleph }_1$.

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• Cofinal types of ultrafilters over measurable cardinals
• Amenable colorings
• Non-Galvin Filters
• Galvin's property at large cardinals and an application to partition calculus
• Magidor-like forcing and the cofinality of the Galvin number