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).
I add a few more references and applications:

> 
> **(A)** Trevor Irwin's thesis ``Fraïssé limits and colimits with applications to continua'':
> 
> 
> 

**Abstract:** The classical Fraïssé construction is a method of taking a direct limit of a family of finite models of a language provided the family fulfills certain amalgamation conditions. The limit is a countable model of the same language which can be characterized by its (injective) homogeneity and universality with respect to the initial family of models. A standard example is the family of finite linear orders for which the Fraïssé limit is the rational numbers with the usual ordering.

We present this classical construction via category theory, and within this context we introduce the dual construction. This respectively constitutes the Fraïssé colimits and limits indicated in the title. We provide several examples.

We then present the projective Fraïssé limit as a special case of the dual construction, and as such it is the categorical dual to the classical (injective) Fraïssé limit. In this dualization we use a notion of model theoretic structure which has a topological ingredient. This results in the countable limit structures being replaced by structures which are zero-dimensional, compact, second countable spaces with the property that the relations are closed and the functions are continuous.

We apply the theory of projective Fraïssé limits to the pseudo-arc by first representing the pseudo-arc as a natural quotient of a projective Fraïssé limit. Using this representation we derive topological properties of the pseudo-arc as consequences of the properties of projective Fraïssé limits. We thereby obtain a new proof of Mioduszewski’s result that the pseudo-arc is surjectively universal among chainable continua, and also a homogeneity theorem for the pseudo-arc which is a strengthening of a result due to Lewis and Smith. We also find a new characterization of the pseudo-arc via the homogeneity property.

We continue with further applications of these methods to a class of continua known as pseudo-solenoids, and achieve analogous results for the universal pseudo-solenoid.

> 
> **(B)** Wieslaw Kubiś, ''Fraisse sequences - a category-theoretic approach to universal homogeneous structures'':
> 
> 
> 

**Abstract:** We present a category-theoretic approach to universal homogeneous objects, with applications in the theory of Banach spaces and in set-theoretic topology,