Question: 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.

Answer: 
If you are willing to allow uncountable generalizations of Fraisse's/Hrushovski construction then the conditions you need essentially are those that make up an [Abstract Elementary Class](http://en.wikipedia.org/wiki/Abstract_elementary_class). Abstract elementary classes are a framework for studying abstract logics introduced by Shelah and generalize many important known examples of logics. As for the relationship to set theory, Will Boney has recently proved from large cardinals the Shelah's categoricity conjecture (see [Tameness](http://homepages.math.uic.edu/~wboney/BoneyTameLC.pdf)). 

If you are willing to generalize just a little further then you get the notion of an [accessible category](http://ncatlab.org/nlab/show/accessible+category). Accessible categories have strong connections to [Vopenka's principle](http://ncatlab.org/nlab/show/Vop%C4%9Bnka's+principle).
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.