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?

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