The Halton sequence and its discrepancy in the Cantor expansion

Let (kappa ) be an infinite cardinal, and (2^kappa . We prove that if there is a weak diamond on (kappa ^+) then every ({C_alpha :alpha 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.