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  1. Complexity Ranks of Countable Models.Su Gao - 2007 - Notre Dame Journal of Formal Logic 48 (1):33-48.
    We define some variations of the Scott rank for countable models and obtain some inequalities involving the ranks. For mono-unary algebras we prove that the game rank of any subtree does not exceed the game rank of the whole model. However, similar questions about linear orders remain unresolved.
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  • Uncountable structures are not classifiable up to bi-embeddability.Filippo Calderoni, Heike Mildenberger & Luca Motto Ros - 2019 - Journal of Mathematical Logic 20 (1):2050001.
    Answering some of the main questions from [L. Motto Ros, The descriptive set-theoretical complexity of the embeddability relation on models of large size, Ann. Pure Appl. Logic164(12) (2013) 1454–1492], we show that whenever κ is a cardinal satisfying κ<κ=κ>ω, then the embeddability relation between κ-sized structures is strongly invariantly universal, and hence complete for (κ-)analytic quasi-orders. We also prove that in the above result we can further restrict our attention to various natural classes of structures, including (generalized) trees, graphs, or (...)
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