The Hart-Shelah example, in stronger logics

We generalize the Hart-Shelah example [10] to higher infinitary logics. We build, for each natural number k≥2 and for each infinite cardinal λ, a sentence ψ_k^λ of the logic L_(2^λ)⁺,ω that (modulo mild set theoretical hypotheses around λ and assuming 2^λ<λ^+m) is categorical in λ⁺,…,λ^+k−1 but not in ℶ_k+1(λ)⁺ (or beyond); we study the dimensional encoding of combinatorics involved in the construction of this sentence and study various model-theoretic properties of the resulting abstract elementary class K^⁎(λ,k)=(Mod(ψ_k^λ),≺_(2^λ)⁺,ω) in the finite interval of cardinals λ,λ⁺,…,λ^+k.