Ryll-Nardzewski Theorem

The Ryll-Nardzewski Theorem, discovered independently in 1959 by Erwin Engeler, Czeslaw Ryll-Nardzewski and Lars Svenonius, characterizes when a complete first-order theory in a countable language is omega-categorical, meaning it has exactly one countably infinite model up to isomorphism. It gives several equivalent conditions for this property, including that the automorphism group of the model has only finitely many orbits on each finite power of the model's domain, and that the theory has only finitely many complete types in each finite number of free variables.