Oligomorphic permutation group
= Oligomorphic permutation group
A permutation group on a set is oligomorphic when its diagonal action on ordered $n$-tuples has finitely many orbits for every positive integer $n$. For a countable model of a complete theory, an oligomorphic automorphism group makes every finite-arity <type space> finite: realized types are dense, and only finitely many are possible because types are constant on orbits.