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.
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.