Write $\kappa=|\mathcal U|$ and define on tuples realizing $\operatorname{tp}(\bar b/\varnothing)$ the definable equivalence relation
$$
E(\bar y,\bar z)\quad\Longleftrightarrow\quad
\forall x\bigl(\phi(x,\bar y)\leftrightarrow\phi(x,\bar z)\bigr).
$$
The $E$-classes correspond exactly to the members of $O(\mathcal D)$, because automorphisms of the <monster model> carry $\bar b$ through all realizations of its type.

There are at most $\kappa$ such tuples, so $|O(\mathcal D)|\leq\kappa$. If the orbit is infinite, recursively choose $\bar b_\alpha$ for $\alpha<\kappa$ in pairwise different $E$-classes. At stage $\alpha$, the type
$$
\operatorname{tp}(\bar b/\varnothing)\cup
\{\neg E(\bar y,\bar b_\beta):\beta<\alpha\}
$$
is finitely satisfiable because there are infinitely many classes, and its parameter set has size less than $\kappa$. Saturation realizes it in $\mathcal U$. Thus there are at least $\kappa$ classes, proving the <orbit of a definable set in a monster model> identity
$$
|O(\mathcal D)|=|\mathcal U|.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 144 / 1 / b / Solution

{title2=$O(\mathcal D)$}

If $\mathcal D=\phi(\mathcal U,b)$, its automorphism orbit consists of the sets $\phi(\mathcal U,f(b))$. If this orbit is infinite, saturation constructs $|\mathcal U|$ pairwise distinct conjugates, so $|O(\mathcal D)|=|\mathcal U|$.


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