An <atomic model> is a model in which the type over the empty set of every finite tuple is isolated. Let $\bar a\mapsto\bar b$ be partial elementary and take $c\in M$. Choose a formula $\theta(\bar x,y)$ isolating $\operatorname{tp}(\bar a,c)$. Since $M\models\exists y\,\theta(\bar a,y)$, elementarity gives $N\models\exists y\,\theta(\bar b,y)$. Any witness $d$ realizes the isolated joint type, so $\bar a,c\mapsto\bar b,d$ is partial elementary. This is the <one-point extension between atomic models>.

Alternately applying this extension property in the two directions to enumerations of countable atomic models constructs a <back-and-forth method>[back-and-forth] isomorphism.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 144 / 4 / i / Solution

If $M,N$ are infinite atomic models of a complete theory and a finite partial elementary map sends $\bar a$ to $\bar b$, then for every $c\in M$ there is $d\in N$ extending it by $c\mapsto d$. An isolating formula for $\operatorname{tp}(\bar a,c)$ supplies $d$. Back-and-forth therefore makes any two countable atomic models isomorphic.


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 Incoming links: One-point extension between atomic models