Let $\mathcal{C}=\{M\_i\}$ be a Fraïssé class of finite $\mathcal{L}$-structures with the generic model $M$. Also, let $M^\*=\prod\_U M\_i$ the be ultraproduct of members of $\mathcal{C}$ where $U$ is a non-principal ultrafilter.


Question. What is the relation between $M$ and $M^\*$?



Fraïssé Limit and Ultraproduct
First note that $M$ is countable while $M^\*$ has size continuum. So the two structures are never isomorphic. But you may instead ask when/if they are elementarily equivalent; and the answer is “sometimes”. 


Example 1: Let $\mathcal{C}$ be the class of finite graphs. Then $M$ is the countable random graph, and one can have that $M^\*$ is elementarily equivalent to $M$ (for example by taking an ultraproduct of Paley graphs). 


Example 2: Let $\mathcal{C}$ be the class of finite linear orders. Then $M$ is isomorphic to $(\mathbb{Q},<)$, while $M^\*$ is a ($\aleph\_1$-saturated) discrete linear order with endpoints. 


EDIT: In the question “an ultraproduct of members of $\mathcal{C}$” is ambiguous, since perhaps you allow one to take an ultraproduct of a subclass. This doesn’t make much difference for Example 2, but in Example 1 one could potentially get different answers for $M^\*$ by varying the subclass (or even the ultrafilter). For example, an ultraproduct of the class of triangle-free graphs will not be a model of the theory of the random graph.


