Fraïssé's famous result on homogeneous countable structures gives us a general method to contruct a countable homogeneous structure starting from the set of its finitely generated substructures. My question is if it is known a method of comparable generality to construct an uncountable homogeneous structure given a class of finitely generated structures.
In other words, I'm looking for something along the line:
Let $\mathcal{C}$ be a set of finitely generated structures satisfying certain properties, with $|\mathcal{C}|\le \aleph_1$. Then there is a structure $K$ of size $\aleph_1$ which is homogeneous and such that $\text{Age}(K)=\mathcal{C}$.
What if we ask $|\mathcal{C}| \le \aleph_0$?
Thanks
Constructing uncountable structures in this way quickly runs into problems. Laskowski/Shelah give an example of an age of size $\aleph_2$ with no limit, homogeneous or not; meanwhile, while ages of size $\aleph_1$ do have limits via an argument building off of the countable case (see Knight/Montalban/Schweber, Lemma 3.11), they need not be homogeneous. If memory serves, Hjorth gave a simplified proof of the Laskowski/Shelah result, but I can't seem to find it at the moment.
The most recent paper on uncountable Fraisse limit constructions I am aware of is Kudaibergenov, but I have not read it so I cannot vouch for its correctness; I mention this only because I have seen multiple sources claim (falsely) that Fraisse's argument extends to uncountable ages, so a bit of care is appropriate here.