Abstract

Let λ denote a singular cardinal. Zeman, improving a previous result of Shelah, proved that $\square _{\lambda}^{\ast}$ together with 2 λ = λ⁺ implies $\lozenge _{S}$ for every S ⊆ λ⁺ that reflects stationarily often. In this paper, for a set S ⊆ λ⁺, a normal subideal of the weak approachability ideal is introduced, and denoted by I[S; λ]. We say that the ideal is fat if it contains a stationary set. It is proved: 1. if I[S; λ] is fat, then $\text{NS}_{\lambda ^{+}}$ ↾ S is non-saturated; 2. if I[S; λ] is fat and 2 λ = λ⁺, then $\lozenge _{S}$ holds; 3. $\square _{\lambda}^{\ast}$ implies that I[S; λ] is fat for every S ⊆ λ⁺ that reflects stationarily often; 4. it is relatively consistent with the existence of a supercompact cardinal that $\square _{\lambda}^{\ast}$ fails, while I[S; λ] is fat for every stationary S ⊆ λ⁺ that reflects stationarily often. The stronger principle $\lozenge _{\lambda ^{+}}^{\ast}$ is studied as well