Abstract
Let \(\kappa \) be an infinite cardinal, and \(2^\kappa <\lambda \le 2^{\kappa ^+}\). We prove that if there is a weak diamond on \(\kappa ^+\) then every \(\{C_\alpha :\alpha <\lambda \}\subseteq \mathcal {D}_{\kappa ^+}\) 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 \(\aleph _1\)). We derive some consequences about weakly inaccessible cardinals. We also prove that the negation of a similar property follows from the proper forcing axiom.
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Garti, S. Weak diamond and Galvin’s property. Period Math Hung 74, 128–136 (2017). https://doi.org/10.1007/s10998-016-0153-0
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DOI: https://doi.org/10.1007/s10998-016-0153-0