= 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.
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