Categoricity,amalgamation, and tameness

Theorem: For each 2 ≤ k < ω there is an -sentence ϕ_k such that (1) ϕ_k is categorical in μ if μ≤ℵ_k−2; (2) ϕ_k is not ℵ_k−2-Galois stable (3) ϕ_k is not categorical in any μ with μ>ℵ_k−2; (4) ϕ_k has the disjoint amalgamation property (5) For k > 2 (a) ϕ_k is (ℵ₀, ℵ_k−3)-tame; indeed, syntactic first-order types determine Galois types over models of cardinality at most ℵ_k−3; (b) ϕ_k is ℵ_m-Galois stable for m ≤ k − 3 (c) ϕ_k is not (ℵ_k−3, ℵ_k−2). The first author is partially supported by NSF grant DMS-0500841.