Journal of Symbolic Logic 67 (1):221 - 234 (2002)

Abstract
We prove that every n-modal logic between K n and S5 n is undecidable, whenever n ≥ 3. We also show that each of these logics is non- finitely axiomatizable, lacks the product finite model property, and there is no algorithm deciding whether a finite frame validates the logic. These results answer several questions of Gabbay and Shehtman. The proofs combine the modal logic technique of Yankov-Fine frame formulas with algebraic logic results of Halmos, Johnson and Monk, and give a reduction of the (undecidable) representation problem of finite relation algebras
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DOI 10.2178/jsl/1190150040
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Hybrid Formulas and Elementarily Generated Modal Logics.Ian Hodkinson - 2006 - Notre Dame Journal of Formal Logic 47 (4):443-478.
Neat Embeddings, Omitting Types, and Interpolation: An Overview.Tarek Sayed Ahmed - 2003 - Notre Dame Journal of Formal Logic 44 (3):157-173.
Omitting Types in Fragments and Extensions of First Order Logic.Tarek Sayed Ahmed - 2021 - Bulletin of the Section of Logic 50 (3):249-287.

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