Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 144 / 3 / b

Let $p(x)\in S(A)$ with $A\subseteq N$ and $|A|<\kappa$. Realize $p$ by $c$ in some elementary extension, and use the Downward Lowenheim-Skolem theorem to choose a model $M_0$ containing $A\cup \{c\}$ with $|M_0|<\kappa$. Universality gives an elementary embedding $e:M_0\to N$. Its restriction sends $A$ elementarily to $e(A)$. Homogeneity extends the inverse partial elementary map $e(A)\to A$ to an automorphism $\sigma$ of $N$. Then $\sigma (e(c))$ realizes $p$ over $A$. Thus the universal homogeneous model is saturated.