Additive and multiplicative Gower's Ramsey theorem

Abstract

W. T. Gower generalized Hindman's Finite sum theorem over $X_k=\left\{\left(n_1,n_2,\dots ,n_k\right):n_1\ne 0\right\}$ by showing that for any finite coloring of $X_k$ there exists a sequence such that the Gower subspace generated by that sequence is monochromatic. For $k=1,$ this immediately gives the finite sum theorem. In this article we will show that for any finite coloring of $X_k$ there exist two sequences $\left\{{\mathbf{n}}_{\mathbf{i}}:i\in I\right\}$ and $\left\{{\mathbf{m}}_{\mathbf{i}}:i\in I\right\}$ such that the Gower subspace generated by $\left\{{\mathbf{n}}_{\mathbf{i}}:i\in I\right\}$ and set of all finite products of $\left\{{\mathbf{m}}_{\mathbf{i}}:i\in I\right\}$ are in a single color. This immediately generalize a result of V. Bergelson and N. Hindman which says that for any finite coloring of $\mathbb{N}$, there exist two sequences ${\left(x_n\right)}_n$ and ${\left(y_n\right)}_n$ such that the finite sum and product generated by ${\left(x_n\right)}_n$ and ${\left(y_n\right)}_n$ are in a same color.