Partition theorems for Abelian groups

Let A be a finite matrix with integral entries and G be an Abelian group. Define A to be partition regular in G if for every partition of G/(0) into finitely many classes there exist elemens x₁,…,x_m contained in one class such that A(x₁,…,x_m)^T = 0. Theorem. A is partition regular in G iff at least one of the following statements holds. (i) There is x ∈ G/(0) such that A(x,…,x)^T = 0. (ii) A is partition regular in Z_p^?₀ (p prime) and Z_p^?₀ ? G. (iii) A is partition regular in Z and the set of orders of elements in G is unbounded.