A generalization of Neumann's Question

Let $G$ be a group, $m\geq2$ and $n\geq1$. We say that $G$ is an $\mathcal{T}(m,n)$-group if for every $m$ subsets $X_1, X_2, \dots, X_m$ of $G$ of cardinality $n$, there exists $i\neq j$ and $x_i \in X_i, x_j \in X_j$ such that $x_ix_j=x_jx_i$. In this paper, we give some examples of finite and infinite non-abelian $\mathcal{T}(m,n)$-groups and we discuss finiteness and commutativity of such groups. We also show solvability length of a solvable $\mathcal{T}(m,n)$-group is bounded in terms of $m$ and $n$.