Let $K$ be a class of finitely generated structures. Fraïssé isolated conditions for which we can define the Fraïssé limit of $K$, a countable structure $M$ in which all structures in $K$ embeds and which is homogeneous in the sense that any isomorphism between finite substructures of $M$ can be extended to an automorphism of $M$. For instance, the order of the rationals is the Fraïssé limit of the class of finite linear orders, the Rado graph is the Fraïssé limit of the class of finite graphs and the atomless countable Boolean algebra is the Fraïssé limit of the class of finite Boolean algebras. The goal of this talk (and probably the next one) will be to give a formulation and a proof of the Fraïssé correspondence for metric structures, which extends the classical Fraïssé theory. We will be mainly following a proof due to Tsankov, which will be the occasion to introduce central notions in (metric) model theory, like metric structures, (quantifier-free) types, type spaces, ultrahomogeneity or existential closedness.
