A relational structure $A$ with an $\omega-$categorical theory $Th(A)$ is ultrahomogenous iff $Th(A)$ admits quantifier elimination. (*)
I was wondering wether the structure $A$ has to be countable...
Think of the following example:
We have the set $A= (0,1)_\mathbb Q \cup (1,2)_\mathbb{R}$ and one binary relation $<$ (the linear order). I think $A$ admits quantifier elimination, is $\omega-$categorical (because the theory is that of a dense linear order) but NOT ultrahomogenous as $\mathbb{Q}$ and $\mathbb{R}$ are not isomorph.
What do you think?
Thank you in advance for any help!
Supplement
I haven’t had much time to do modeltheory the last weeks, but today I tried to proof the other direction of *. And I would appreciate some help again!
Claim: If a countable structure $A$ has an $\omega$-categorical theory and admits QE, then it is ultrahomogeneous.
Proof: Assume $Th(A)$ admits quantifier elimination, $\bar{a}, \bar{b} \in A^n$ and $ f: \bar{a} \mapsto \bar{b}$ is an isomorphism. We have to show: there exists an automorphism $f’$ so that $f'(\bar{a}) = \bar{b}$.
As $Th(A)$ is $\omega-$categorical and admits QE, there exists a quantifier-free formular $\varphi(\bar{x})$, that isolates tp($\bar{a}$). And because of the isomorphism $f$ we get: $A \models \varphi( \bar{b})$, hence tp($\bar{a}$)=tp($\bar{b}$). At this point I don't know how to argue, that whenever two n-tuples have the same complete type, there exists an automorphism between them. Do you have an idea? Thank you!