Specializing Aronszajn trees and preserving certain weak diamonds

by Mildenberger and Shelah. [MdSh:848]
J Applied Analysis, 2009
We show that diamondsuit (R, L, not in) together with ``all Aronszajn trees are special'' is consistent relative to ZFC. The weak diamond for the uniformity of Lebegue null sets was the only weak diamond in the Cichon diagramme for relations whose consistency together with ``all Aronszajn trees are special'' was not yet settled. We can have CH or 2^{aleph_0} = aleph_2 . Our techniques give more on coverings by related small sets that are preserved in iterations that are stronger relatives to [Sh:f, Chap. V, sect. 5--7]

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