Cofinalities of Elementary Substructures of Structures on א

The question is whether Sf is stationary in [αω] <אn∗+1 . The question was first posed by Baumgartner in [B7]. By a standard result, the above question can also be rephrased as certain transfer property. Namely, Sf is stationary iff for any structure A = 〈אω, . . . 〉 there’s a B ≺ A such that |B| = אn∗ and for all n ∈ X we have cf(B ∩ אn) = אf(n). Note that if f is eventually constant then Sf is always stationary. (see [B7]). Also, any reasonable “finite variation” of f will preserve the property, i.e., if n1 > n ∗ and f1 : X → n ∗ + 1 agrees with f on dom(f)\(n1 + 1), then Sf1 is stationary provided that Sf is. So we are interested in the case that f is not eventually constant. You may wonder how strong the statement that Sf is stationary is. Magidor (unpublished, but close to [Mg3]) has shown that if f : ω\2 → 2 assumes the values 0 and 1 alternatively and Sf