Counting siblings in universal theories

TL;DR

This paper demonstrates that non-cellular countable structures in finite relational languages can be extended to uncountably many bi-embeddable structures, highlighting a deep connection between structure properties and their extensions.

Contribution

It establishes a new link between the cellularity of structures and the abundance of bi-embeddable extensions in the context of countable models.

Findings

Non-cellular structures have uncountably many bi-embeddable extensions.

The proof uses a case division based on mutual algebraicity.

Extensions preserve the age of the original structure.

Abstract

We show that if a countable structure $M$ in a finite relational language is not cellular, then there is an age-preserving $N \supseteq M$ such that $2^{\aleph_0}$ many structures are bi-embeddable with $N$. The proof proceeds by a case division based on mutual algebraicity.