I have a question about the following paper:


**One-dimensional asymptotic classes of finite structures**


by Macpherson and Steinhorn [(link at Trans. AMS website)](https://www.ams.org/journals/tran/2008-360-01/S0002-9947-07-04382-6/S0002-9947-07-04382-6.pdf).




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Let $\mathbf{K}$ be a one-dimensional asymptotic class of finite structures.


The phrase "**every infinite ultraproduct**" appear in this paper several times. For example Lemma 2.5. This phrase is vague for me. What does that mean? Does it mean that the ultraproduct of evey infinite subclass of $\mathbf{K}$? or it means the ultraproduct of **all** members of $\mathbf{K}$ under different ultrafilters? 



On asymptotic classes of finite structures
At risk of triviality: An infinite ultraproduct (of members of $\mathcal{C}$) is a structure which is (a) infinite, and (b) an ultraproduct (of members of $\mathcal{C}$).


So, for example, Lemma 2.5 reads: "Let $\mathcal{C}$ be a class of finite $\mathcal{L}$-structures, and suppose that every infinite ultraproduct of members of $\mathcal{C}$ is strongly minimal. Then $\mathcal{C}$ is a $1$-dimensional asymptotic class."


This could be equivalently stated as follows: "Let $\mathcal{C}$ be a class of finite $\mathcal{L}$-structures. Suppose that for every family $\{A\_i\}\_{i \in I}$ of members of $\mathcal{C}$, and every ultrafilter $\mathcal{U}$ on $I$, if the ultraproduct $M = \prod\_{i\in I} A\_i\,/\, \mathcal{U}$ is infinite, then $M$ is strongly minimal. Then $\mathcal{C}$ is a $1$-dimensional asymptotic class." 


Of course, if you take the ultraproduct of a family of finite structures by a principal ultrafilter, the result will be a finite structure. The point of writing "infinite ultraproduct" is to rule out such trivial cases. 


