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 R on the generalized Cantor space 2κ there is an Lκ+κ-sentence φ such that the embeddability relation on its models of size κ, which are all trees, is Borel bi-reducible (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 on 2κ. These facts generalize analogous results for κ=ω obtained in Louveau and Rosendal (2005) [17] and Friedman and Motto Ros (2011) [6], and it also partially extends a result from Baumgartner (1976) [3] concerning the structure of the embeddability relation on linear orders of size κ.