An application of ramsey's theorem to groups with homogeneous theory

For any integers n >m ≥ 2, we say that a complete theory T is (m, n)-homogeneous if, for each model M of T, two n-tuples ⱥ,ƀ in M have the same type if the corresponding m-tuples from ⱥ and ƀ have the same type. It was conjectured by H. Kikyo that, if M is an infinite group, with possibly additional structure, then the theory of M is not (m, n)-homogeneous. We prove a general result on structures with (m, n)-homogeneous theory which implies that, if M is a counterexample to this conjecture, then there exists an integer h such that each abelian subgroup of M has at most h elements. It follows that there exist an integer k such that M k = 1, and an integer l such that each finite subgroup of M has at most l elements.