Diamond, GCH and weak square

Abstract

Shelah proved recently that if κ > ω and S ⊂ κ + is a stationary set of ordinals of cofinality different from cf(κ), then 2 κ = κ + implies ◇ κ+ (S). We show that for singular κ, an elaboration on his argument allows us to derive ◇ κ+ (T) from 2 κ = κ + + □ * κ where T = {δ < κ + | cf(δ) = cf(κ)}. This gives a strong restriction on the existence of saturated ideals on κ + .