# Omega-categorical theory without quantifier elimination

Is there an $$\omega$$-categorical theory without quantifier elimination? The way I normally prove $$\omega$$-categoricity (with back-and-forth) immediately gives me QE as a corollary of the test Theorem 13.7 (see picture). My definition of local isomorphism is: • You should probably include the definition of "local isomorphism" in your question as well, because I do not think this is really a standard definition. At least, I think it is easily confused with a partial elementary map (which it is not). Jun 20, 2019 at 15:58
• @MarkKamsma I've added my working definition. What is your definition? Jun 20, 2019 at 16:57
• Personally I know these notes, so I knew what this definition was. It's just that most people might expect it to mean "partial elementary embedding", which is the same definition but then it should preserve all formulas. This is really something different: we can always extend a partial elementary embedding (on a set of cardinality $<\kappa$) into a $\kappa$-saturated structure by one more element (exercise!) Jun 20, 2019 at 17:01
• @MarkKamsma The exercise is a lemma in the notes :) . If I had not known it by heart I would surely be a good exercise, so thanks! I didn't know Benno's notes were in use in Utrecht. But now that I think about it, it makes a lot of sense. Jun 20, 2019 at 18:29
• They weren't, I just followed a few courses at the UvA ;) Jun 20, 2019 at 18:51

One example is the theory of an equivalence relation with infinitely many classes of size $$1$$ and infinitely many classes of size $$2$$.
• @AlexKruckman Theorem. Let $T$ be omega-categorical and let $M$ be the countable model of $T$. $M$ is homogeneous iff $Th(M)$ has Q.E. An infinite graph with a single edge is homogeneous, isn't it? Jun 8, 2021 at 7:16
• @Alice.H No, it's not homogeneous. Let $v$ be an endpoint of the single edge, and let $w$ be any element which is not an endpoint of the single edge. Then the substructures $\{v\}$ and $\{w\}$ are isomorphic, but there is no automorphism of the model moving $v$ to $w$. Jun 8, 2021 at 12:05