A permutability problem in infinite groups and Ramsey's theorem

Abstract

We use Ramsey's theorem to generalise a result of L. Babai and T.S. Sós on Sidon subsets and then use this to prove that for an integer n > 1 the class of groups in which every infinite subset contains a rewritable n-subset coincides with the class of groups in which every infinite subset contains n mutually disjoint non-empty subsets X1, ..., Xn such that X1 ⋯ Xn ∩ Xσ(1) ⋯ Xσ(n) ≠ 0 for some non-identity permutation σ on the set {1, ..., n}.