An Application of Ramsey’s Theorem to Groups with Homogeneous Theory

Francis Oger

We investigate the following conjecture,which was proposed by H. Kikyo in 1996:

Definition. For any integers n > m ≥ 2, a complete theory T is (m, n)-homogeneous if, for each model M of T, two n-tuples \overline{a}, \overline{b} in M have the same type if the corresponding m-tuples from \overline{a} and \overline{b} have the same type.

Conjecture. If M is an infinite group, with possibly additional structure, then, for any integers n > m ≥ 2, the theory of M is not (m, n)-homogeneous.

We proved the theorem below, using a variant of Ramsey’s Theorem which follows from more general results by R.L. Graham, K. Leeb and B.L. Rotschild:

Theorem. Let L be a language, let M be an L-structure, and let ∗(u, v, w) be a parameter-free Lformula. If ∗ defines an abelian group structure on an infinite subset or on arbitrarily large finite subsets of M, then the theory of M is not (m, n)-homogeneous for n > m ≥ 2.

Consequently, if M is a counterexample to Kikyo’s 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. The group M necessarily has infinite finitely generated subgroups, and any such subgroup is a counterexample to Burnside’s conjecture.

Address: Equipe de Logique Mathématique, Université Paris VII - C.N.R.S., 2 place Jussieu, case 7012, 75251 Paris Cédex 05, FRANCE

E-mail: oger@logique.jussieu.fr