Uncountable structures are not classifiable up to bi-embeddability

Answering some of the main questions from [MR13], 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 groups. This fully generalizes to the uncountable case the main results of [LR05,FMR11,Wil14,CMR17].

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