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Let $M$ be a countable structure that is homogeneous, i.e. every isomorphism between finitely generated substructures of $M$ extends to an automorphism of $M$. Let $\phi_M$ be the Scott sentence of $M$.

If $N$ is an uncountable model of $\phi_M$, then $N$ is not necessarily homogeneous, but is the existence of an uncountable model sufficient for the existence of an uncountable homogeneous model? If so, can we further conclude that given $N\models\phi_M$, $N$ uncountable, there is a homogenous model of $\phi_M$ of the same cardinality as $N$?

Any help?

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@Ioannis: there is an interesting special case of the question you are asking, when M is a countable recursively saturated model of $PA$, for which I suspect that the answer to your first question is negative. If I make any progress, I will let you know. –  Ali Enayat Jun 19 '11 at 17:41
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@Ioannis: Update, it turns out that the answer to your question is POSITIVE for the recursively saturated $M$; which you may have already known about, if not, see Theorem 10 of: S. Buechler, Steven, Expansions of models of $\omega $ω-stable theories. J. Symbolic Logic 49 (1984), no. 2, 470–477 –  Ali Enayat Jun 21 '11 at 14:24
    
Thank you for the reference. I will have to go through it to clarify all the details. It seems to me a natural question to ask. Someone might have thought about it already even in the non-recursive case. –  Ioannis Souldatos Jun 21 '11 at 16:12

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