Weak diamond and Galvin’s property

Let κ be an infinite cardinal, and 2κ<λ≤2κ+. We prove that if there is a weak diamond on κ⁺ then every {Cα:α<λ}⊆Dκ+ satisfies Galvin’s property. On the other hand, Galvin’s property is consistent with the failure of the weak diamond (and even with Martin’s axiom in the case of ℵ₁). We derive some consequences about weakly inaccessible cardinals. We also prove that the negation of a similar property follows from the proper forcing axiom.