Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 144 / 4 / b

Let $\kappa =|N|$. If the saturated model $N$ had a basis $B$ of cardinality below $\kappa$, then $N=\mathrm{acl}(B)$ would omit the consistent generic type in a strongly minimal theory over $B$. This contradicts $\kappa$-saturation. A basis is a subset of $N$, so its cardinality is at most $\kappa$. Therefore

Assume $\dim (N)=|N|=\kappa$, with $\kappa \ge |L|+{\aleph }_0$. Let $A\subseteq N$ have size below $\kappa$ and let $p(x)\in S_1(A)$ be a complete type. If $p$ is algebraic, its realizations lie in $\mathrm{acl}(A)\subseteq \mathrm{acl}(N)=N$. If $p$ is nonalgebraic, strong minimality makes it the unique generic type over $\mathrm{acl}(A)$. The pregeometry exchange property givesso some basis element of $N$ lies outside $\mathrm{acl}(A)$ and realizes $p$.

Thus $N$ realizes every one-type over every parameter set of size below $\kappa$. The standard one-type characterization, applied successively to tuple coordinates, makes $N$ $\kappa$-saturated. Consequently