Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 144 / 8 / ii / Solution

A formula $\varphi (\bar{x},\bar{y})$ has the order property for $T$ when, for every finite $n$, some model contains tuples $a_i,b_j$ withCompactness realizes this pattern indexed by any linear order.

Fix $\kappa \ge |T|$ and let $\lambda$ be least with $2^{\lambda }>\kappa$. Use the order property along $I={\mathbb{Q}}^{\lambda }$, and take parameterswhere part i gives $|B|\le \kappa$. For distinct $i,i^{\prime }\in I$, choose $j\in J$ strictly between them. Then $\varphi (a_i,b_j)$ and $\varphi (a_{i^{\prime }},b_j)$ have different truth values, so the types $\mathrm{tp}(a_i/B)$ are distinct. There are $|I|=2^{\lambda }>\kappa$ such types over at most $\kappa$ parameters. Enlarging $B$ to size exactly $\kappa$ if necessary preserves them. Hence $T$ is not $\kappa$-stable. This is the order property implies instability argument.