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An injective map between subsets of structures is partial elementary when it preserves every first-order formula on tuples from its domain. This is stronger than a partial embedding, which only preserves quantifier-free formulas. In countable models, successive one-point extensions in both directions produce an elementary isomorphism by the back-and-forth method.

The theory of the random graph has the extension property: for finite disjoint vertex sets $A,B$, there is a new vertex adjacent to every point of $A$ and to no point of $B$.

Let $M,N$ be countable models and let $f_0$ be a finite partial embedding. Enumerate $M=(m_i{)}_{i<\omega }$ and $N=(n_i{)}_{i<\omega }$. At an even stage, take the first $m_i$ outside the domain. Its adjacency pattern to the finite domain prescribes finite disjoint subsets of the range; the extension property in $N$ supplies a new image with exactly that pattern. At an odd stage, apply the same argument to the inverse map and the first unused $n_i$. This back-and-forth method produces an increasing sequence of finite partial embeddings whose union is an isomorphism. Hence every finite partial embedding extends to an isomorphism $M\cong N$.

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.