oligomorphic permutation group

A permutation group acting on a countablyinfinite set is called oligomorphic,if it has finitely many orbits of $n$-tuples,for all $n\ge 1$.

Ryll-Nardzewski, Engeler, and Svenonius proved thata countably infinite first-order structure has an oligomorphicautomorphism group if and only if the structure is$\omega$-categorical.