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: 
A Fraïssé construction lies at the heart of the proof of my embedding theorems.

**Theorem.** (J. D. Hamkins, [Every countable model of set theory
embeds into its own constructible universe](http://jdh.hamkins.org/every-model-embeds-into-own-constructible-universe/), JML 13(2), 2014).

1. For any two countable models of set theory $\langle
M,\in^M\rangle$ and $\langle N,\in^N\rangle$, one of them embeds
into the other.
2. Indeed, a countable model of set theory $\langle M,{\in^M}\rangle$ embeds into another model of set theory $\langle
N,{\in^N}\rangle$ if and only if the ordinals of $M$ order-embed
into the ordinals of $N$.
3. Consequently, every countable model $\langle M,\in^M\rangle$ of
set theory embeds into its own constructible universe $\langle
L^M,\in^M\rangle$.
4. Furthermore, every countable model of set theory embeds into
the hereditary finite sets $\langle \text{HF},{\in}\rangle^M$ of
any nonstandard model of arithmetic $M\models\text{PA}$. Indeed,
$\text{HF}^M$ is universal for all countable acyclic binary
relations.

The sense of embedding here is $j:M\to N$ for which $x\in y\iff j(x)\in j(y)$, which is the model-theoretic sense of embedding, and this is weaker than those usually considered in set theory because they need not be elementary nor even
$\Delta\_0$-elementary. An embedding $j:M\to N$ is simply an isomorphism
of $M$ with a substructure of $N$, not necessarily transitive.

The proof proceeds by finding sufficiently universal substructures
of any model of set theory, using essentially a Fraïssé limit
construction, as you can see in the paper. Basically, this construction shows that every
countable model of set theory $\langle M,\in^M\rangle$ has a
submodel that is universal for all countable $\text{Ord}^M$-graded
digraphs. A *grading* of a digraph is a linear pre-order on that
digraph, such that the linear order strictly increases when
following any edge of the digraph. Any model of set theory
$\langle M,\in^M\rangle$ admits an $\text{Ord}^M$ grading by means
of the von Neumann rank, since whenever $x\in y$, then the rank of
$y$ strictly exceeds that of $x$.

It happens that I am speaking today on this topic at the [CUNY set
theory seminar](http://jdh.hamkins.org/on-embeddings-of-the-universe-into-the-constructible-universe-set-theory-seminar-march-2015/), in about an hour.
While not strictly set theory, but rather a result connecting Ramsey theory and topological dynamics, two areas that are rather close to set theory, you might find the following example interesting:

In <http://arxiv.org/pdf/math/0305241.pdf> Kechris, Pestov and Todorcevic showed that a Fraisse class of finite structure is Ramsey iff the automorphism group of the Fraisse limit is extremely amenable.

Let me explain: The Fraisse limits that we are talking about here are countable structures and the automorphism groups of these structures carry a natural topology, the topology of pointwise convergence, which in this situation is separable and metric. Given a topological group $G$, a continuous action of $G$ on a compact space $X$
is minimal if every orbit is dense in $X$. We call a compact space $X$ together with a continuous $G$-action a $G$-flow. Using Zorn's lemma, we see that every $G$-flow contains a minimal $G$-flow. It turns out that there is a universal minimal $G$-flow, i.e., one that maps onto every other minimal $G$-flow by a $G$-equivariant map (a map that respects the group action). Now the construction of the universal minimal flow can be phrased in various ways, but concepts like ultrafilters and other applications of Zorn's lemma show up here.
So this is, in some sense, set theory. 

Now a topological group is extremely amenable if its universal minimal flow has just one point, or equivalently, if every $G$-flow has a fixed point.

On the other hand, a class of finite structures is Ramsey if for all $A$ and $B$ in the class there is $C$ in the class such that for every coloring of all copies of $A$ inside $C$ with two colors, there is a copy $B'$ of $B$ in $C$ such that all copies of $A$ in $B'$ have the same color. The usual Ramsey theorem shows that the class of finite linear orders is a Ramsey class and the Nesetril-Rödl theorem is precisely the statement that the class of finite ordered graphs is a Ramsey class. 

As corollaries from the Kechris, Pestov, Todorcevic theorem we get that the automorphism groups of the countable dense linear order without endpoint $(\mathbb Q,\le)$ and of the ordered random graph are extremely amenable.