The descriptive set-theoretical complexity of the embeddability relation on models of large size

We show that if κ is a weakly compact cardinal then the embeddability relation on (generalized) trees of size κ is invariantly universal. This means that for every analytic quasi-order on the generalized Cantor space 2 there is an Lκ+κsentence φ such that the embeddability relation on its models of size κ, which are all trees, is Borel bireducible (and, in fact, classwise Borel isomorphic) to R. In particular, this implies that the relation of embeddability on trees of size κ is complete for analytic quasi-orders. These facts generalize analogous results for κ = ω obtained in [LR05, FMR11], and it also partially extends a result from [Bau76] concerning the structure of the embeddability relation on linear orders of size κ.