Uncountable homogeneous structures
Adam Bartoš, Wiesław Kubiś
TL;DR
The paper addresses the problem of existence for uncountable homogeneous first‑order structures with a given age, introducing a general construction that starts from a countable Fraïssé limit ${\mathbb U}$ with a nontrivial self‑embedding and builds an uncountable chain of copies whose colimit remains homogeneous and retains the same age. The core idea hinges on extensible embeddings and the automorphism monoid ${\mathcal{E}{\mathbb U}}$, using either natural Katětov‑style constructions or careful amalgamation arguments to ensure homogeneity at the uncountable level. It presents a unifying framework that encompasses ultraproducts, Katětov functors, and rigid moieties, and analyzes when the amalgamation property lifts to infinite structures, illustrated by diverse examples (groups, anti‑metric spaces, and ordered graphs). The paper also discusses converse directions, poses open problems about extensibility in various Fraïssé limits, and suggests a broader categorical perspective via $\omega$‑accessible categories. Overall, it broadens Fraïssé theory to uncountable contexts, clarifies the role of extensibility and AP, and provides concrete constructions and counterexamples across multiple domains.
Abstract
We study the existence of uncountable first-order structures that are homogeneous with respect to their finitely generated substructures. In many classical cases this is either well-known or follows from general facts, for example, if the language is finite and relational then ultrapowers provide arbitrarily large such sturctures. On the other hand, there are no general results saying that uncountable homogeneous structures with a given age exist. We examine the monoid of self-embeddings of a fixed countable homogeneous structure and, using abstract Fraïssé theory, we present a method of constructing an uncountable homogeneous structure, based on the amalgamation property of this monoid.