I would like to show that every model complete theory $T$ has an AE axiomatization. The problem comes up as Exercise 2.5.15 in Marker's book, but I'm really not sure how to get started with it
2026-04-01 11:16:51.1775042211
Model complete theories have AE axiomatization
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Exercise 2.5.15 in Marker actually says something different: if T is any theory with the property that the union of any chain of models of T is a model of T, then T has AE axiomatization. (There are hints in the exercise as to how to prove this.) This proves your question, though, because if T is model complete, then any chain is an elementary chain, so the union of a chain of models of T will be a model of T.