TITLE: Application of Fraïssé construction in set theory
QUESTION [15 upvotes]: As you know Fraïssé limit construction and its generalization, Hrushovski's construction, have many applications in model theory to build models with interesting property.
Now I would like to know the application of Fraïssé construction in set theory.
Question: What are the major applications of Fraïssé construction in set theory?
Any reference will be appreciated.

REPLY [7 votes]: This is a comment on something Nate Ackerman posted two days ago in the context of Shelah's categoricity conjecture for Abstract Elementary Classes.  As I according to math overflow I don't have the necessary credentials to comment directly, I encloed this.
At present even the consistency of that conjecture is open.  The best ZFC result is for AECs that satisfy extra conditions assuming that the class is categorical in a successor (Grossberg & VanDieren).  Boney managed to derive the above "extra conditions" from a class-many strongly compact cardinals.  In all approximations in the last 25 years to the categoricity conjecture for AECs one had to assume categoricity in \lambda^+ removal for the successor assumptionis considered one of teh major open problems.
Last week Sebastien Vasey posted on the arXiv a rather long paper, his Theorem 1.7(2) establishes the consistency of the eventual categoricity conjecture (removing the successor).  In his proof he is using several major recent results (total of about 500-600 pages) that I am sure are correct. However he is also using a Theorem that Shelah announced several years ago, the draft of that paper of Shelah is more than 150 pages long and he is working on it several years now.