# Uncountable structures are not classifiable up to bi-embeddability

## TL;DR

This paper demonstrates that for uncountable structures of certain sizes, the embeddability relation is highly complex and universal, extending known results from countable to uncountable cases across various classes of structures.

## Contribution

It generalizes the classification complexity of embeddability relations to uncountable structures, showing they are invariantly universal for a broad class of structures.

## Key findings

- Embeddability relations are strongly invariantly universal for uncountable structures.
- Results apply to classes like trees, graphs, and groups.
- Extends countable case results to uncountable structures.

## Abstract

Answering some of the main questions from [MR13], we show that whenever $\kappa$ is a cardinal satisfying $\kappa^{< \kappa} = \kappa > \omega$, then the embeddability relation between $\kappa$-sized structures is strongly invariantly universal, and hence complete for ($\kappa$-)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].