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

TL;DR

This paper demonstrates that the embeddability relation on large trees of size  is as complex as any analytic quasi-order, generalizing previous results from countable to uncountable models under certain set-theoretic assumptions.

Contribution

It establishes the invariant universality and completeness of the embeddability relation on large trees, extending known results from countable models to uncountable models of size .

Findings

Embeddability on -sized trees is invariantly universal.

Embeddability relation is complete for analytic quasi-orders.

Generalizes previous countable case results to uncountable models.

Abstract

We show that if \kappa\ is a weakly compact cardinal then the embeddability relation on (generalized) trees of size \kappa\ is invariantly universal. This means that for every analytic quasi-order R on the generalized Cantor space 2^\kappa\ there is an L_{\kappa^+ \kappa}-sentence \phi\ such that the embeddability relation on its models of size \kappa, 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 \kappa\ is complete for analytic quasi-orders. These facts generalize analogous results for \kappa=\omega\ 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 \kappa.