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

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 $\mathrm{tp}(\bar{a},c)$. 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 isomorphism.