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 $k\ge 2$ and for each infinite cardinal $\lambda$, a sentence ${\psi }_k^{\lambda }$ of the logic $L_{(2^{\lambda }{)}^{+},\omega }$ that (modulo mild set theoretical hypotheses around $\lambda$ and assuming $2^{\lambda }<{\lambda }^{+m}$) is categorical in ${\lambda }^{+},\dots ,{\lambda }^{+k-1}$ but not in ${\mathrm{\beth }}_{k+1}(\lambda {)}^{+}$ (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 ${\mathcal K}^*(\lambda,k)=(Mod(\psi_k^\lambda),\prec_{(2^\lambda)^+,\omega})$ in the finite interval of cardinals $\lambda ,{\lambda }^{+},\dots ,{\lambda }^{+k}$.