Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 144 / 2 / a / Solution

Let $p:A\to \mathcal{U}$ be a small partial embedding. To add a vertex $c$, prescribe for its image adjacency to $p(a)$ exactly when $c$ is adjacent to $a$, together with inequalities excluding the existing image. Every finite part of this prescription is realized by the extension axioms of the theory of the random graph, and saturation realizes the whole type. The same argument applies in the reverse direction. A transfinite back-and-forth method therefore extends $p$ to an automorphism of $\mathcal{U}$. Automorphisms preserve every first-order formula, so $p$ is elementary.