Weak diamond and Galvin's property

TL;DR

This paper explores the relationship between the weak diamond principle and Galvin's property, demonstrating implications, consistency results, and connections to forcing axioms in set theory.

Contribution

It proves that the weak diamond implies Galvin's property and shows their independence under various set-theoretic assumptions.

Findings

Weak diamond implies Galvin's property.

Galvin's property is consistent with negation of weak diamond.

Proper forcing axiom affects Galvin's property at .

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 $\aleph_1$.