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Definition. 1) A complete first-order $\mathcal{L}$-theory $T$ is said to be simple if each type does not fork over some subset $A$ of its domain where $|A|\leq |T|$.

  1. An $\mathcal{L}$-structure $\mathcal{M}$ is called simple if its theory is simple.

Question. Let $\{\mathcal{M}_i\}_{i<\omega}$ be a family of simple $\mathcal{L}$-structures. What can we say about the simplicity of the ultraproduct $\mathcal{M}=\prod_\mathcal{U} \mathcal{M}_i$, where $\mathcal{U}$ is a non-principal ultrafilter on $\omega$?

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In general this is hard. Finite linear orders would satisfy stability and hence be simple. However a non-principal ultraproduct of finite linear orders (chosen to avoid trivialities) would have the strict order property and hence would not be simple.

If you want to follow up, here are some notes from Dario Garcia: http://www1.maths.leeds.ac.uk/~pmtdg/NotesIPM.pdf .

They should give you an idea of what can be said about ultraproducts of finite structures and what sort of questions are currently being considered in that area.

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  • $\begingroup$ Thanks for the shout out! $\endgroup$
    – Darío G
    Commented Oct 18, 2022 at 2:17

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