Keywords: Relational structures, ages, counting functions, oligomorphic groups, age algebra, Ramsey theorem, well quasi ordering, cellular graphs, tournaments.
Groups for which [[theta].sub.G](n) is always finite are said oligomorphic by P.J.Cameron.
Then G is oligomorphic if and only if the complete theory of R is [[??].sub.0]-categorical.
At this point, enough to know that the kernel of any relational structure which encodes an oligomorphic permutation group is finite (this fact immediate: if R encodes a permutation groupGacting on a set E then K(R) is the set union of the orbits of the 1-element subsets of E which are finite.
The orbital profile of an oligomorphic group is either polynomial or faster than every polynomial.