Generalized symmetric systems and thin-very tall compact scattered spaces

We solve a well--known problem in the theory of compact scattered spaces and superatomic boolean algebras by showing that, under GCH and for each regular cardinal $κ\geq ω$, there is a poset $\mathcal P_κ$ preserving all cardinals and forcing the existence of a $κ$--thin very tall locally compact scattered space. For $κ> ω$, we conceive the poset $\mathcal P_κ$ as a higher analogue of the poset $\mathcal P_ω$ originally introduced by Asperó and Bagaria in the context of an (unpublished) alternative consistency proof.

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