= ω of κ-categoricity, and omega-categorical theories are also referred to as ω-categorical.
The notion is most important for countable first-order theories.
In 1959 Erwin Engeler, Czesław Ryll-Nardzewski and Lars Svenonius, proved several independently.
[1] Despite this, the literature still widely refers to the Ryll-Nardzewski theorem as a name for these conditions.
[2][3] Given a countable complete first-order theory T with infinite models, the following are equivalent: The theory of any countably infinite structure which is homogeneous over a finite relational language is omega-categorical.