The Hart-Shelah example, in stronger logics

We generalize the Hart-Shelah example \cite{HaSh:323} to higher infinitary logics. We build, for each natural number $k\geq 2$ 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 $λ^+,\dots,λ^{+k-1}$ but not in $\beth_{k+1}(λ)^+$ (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 ${\mathcal K}^*(λ,k)=(Mod(ψ_k^λ),\prec_{(2^λ)^+,ω})$ in the finite interval of cardinals $λ,λ^+,\dots,λ^{+k}$.