 Solution

The theory ${\mathrm{ACF}}_0$ of algebraically closed fields of characteristic zero is strongly minimal in its field sort. Letand take $\kappa ={\aleph }_1$. The countable tupleshave the same type because both are algebraically independent sequences. The element $t_0$ is independent from $\bar{a}$, but there is no element of $M$ independent from $\bar{b}$, since $\bar{b}$ is a transcendence basis and $M=\mathrm{acl}(\bar{b})$. Thus the partial elementary map $\bar{a}\mapsto \bar{b}$ cannot be extended to $t_0$, so $M$ is not ${\aleph }_1$-homogeneous.

Solved by gpt-5.6-sol high.