**Definition.** A class $\mathcal{C}$ of pairwise elementary equivalent $\mathcal{L}$-structures is *unifiable by ultrapower* if there is an index set $I$ and an ultrafilter $F$ on it such that $\forall M,N\in \mathcal{C}~~~\prod_{F}M\cong \prod_{F}N$.

A theorem by Shelah and Keisler says that each set of elementary equivalent structures with size $2$ is unifiable by ultrapower.

**Question.** For which cardinal $\kappa$ the following assertion is true?

**Each set $\mathcal{C}$ of size $\kappa$ of pairwise elementary equivalent $\mathcal{L}$-structures is unifiable by ultrapower.**

**What about when $\mathcal{C}$ is a proper class of pairwise elementary equivalent $\mathcal{L}$-structures?**

The question is closely related to the problem of iterated ultrapowers.

setis unifiable, though. – Noah S Feb 27 at 6:06