What are some applications of projective Fraïssé limits? I am looking for some applications of projective Fraïssé limits.
For example are they related to a theorem in set theory or topology?
Also is there any modified version for them? (like the version of Hrushovski for Fraïssé limits which led to counterexample to Zilber's conjecture?)
Would you please explain some about their importance?
 A: Before I provide some specific examples of applications, let me just say that the way I like to think of Fraïssé theory (projective or not) is as a convenient framework for translating dynamical properties of Polish groups to combinatorial properties of finite structures and vice versa. See, for example, a survey of Kechris on this.
From this point of view, projective Fraïssé theory allows us to construct natural combinatorial models (which we usually call prespaces) for analyzing the dynamical properties of homeomorphism groups of various compacta.
To mention some concrete examples, projective Fraïssé theory has been used:
(1) in the proof that the homeomorphism group of the Cantor space has ample generics;
(2) in the computation of the universal minimal flow of the Lelek fan;
(3) in providing a new combinatorial proof of the point homogeneity of the Menger sponge;
Projective Fraïssé theory has also been used for providing "canonical" definitions/characterizations of classical continua such as the pseudo-arc and the n-dimensinal Menger compacta. For example, since 1926, there have been numerous definitions of the Menger sponge as an inverse limit of various inverse system of polyhedra. When you read either of these definitions you have the feeling that these inverse systems are somewhat ad hoc, and just by chance they end up giving the same inverse limit. In my opinion, the projective Fraïssé definition of the Menger sponge justifies, on a philosophical level, why the Menger sponge is such a canonical object: it is the inverse limit of the generic inverse system in the category of connected graphs.
Finally, I know of at least two instances where projective Fraïssé theory has been used for introducing entirely new natural compacta whose properties have not been well-studied yet:
(1) In the last Section of this paper a "homology" version of the Hilbert cube is introduced. In fact, more generally a "homology" version of the n-dimensional Menger continuum is introduced for all n>1.
(2) In this paper they introduce a new class of compacta (fences) which are associated to certain classes of finite posets (Hasse posets)

Let me close with an open problem from geometric group theory which, in my opinion, should admit a nice projective Fraïssé approach:
Problem. Characterize all countable groups which admit a faithful action by homeomorphisms on the Menger sponge.
