Monster model ($\mathcal{U}$)

A monster model is a fixed sufficiently saturated and strongly homogeneous model of a complete theory, of a cardinality much larger than every parameter set under discussion.

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