Fraïssé limit via forcing - Mohammad Golshani (Journal of Mahani Mathematical Research)
Suppose $\mathcal{L}$ is a finite relational language and $\mathcal{K}$ is a class of finite $\mathcal{L}$-structures closed under substructures and isomorphisms. It is called aFraïssé class if it satisfies Joint Embedding Property (JEP) and Amalgamation Property (AP). A Fraïssé limit, denoted $Flim(\mathcal{K})$, of aFraïssé class $\mathcal{K}$ is the unique\footnote{The existence and uniqueness follows from Fraïssé's theorem, See \cite{hodges}.} countable ultrahomogeneous (every isomorphism of finitely-generated substructures extends to an automorphism of $Flim(\mathcal{K})$) structure into which every member of $\mathcal{K}$ embeds.Given a Fraïssé class K and an infinite cardinal κ, we define a forcing notion which adds a structure of size κ using elements of K, which extends the Fraïssé construction in the case κ=ω.