Additive and multiplicative Gower's Ramsey theorem

TL;DR

This paper extends Gower's Ramsey theorem by demonstrating the existence of two sequences whose generated subspaces and finite products are monochromatic under any finite coloring, generalizing prior results on sums and products.

Contribution

It introduces a new combinatorial result linking additive and multiplicative structures in colored sets, generalizing previous theorems by Gower, Bergelson, and Hindman.

Findings

Existence of two sequences with monochromatic subspace and finite products

Generalization of Gower's theorem to additive and multiplicative structures

Extension of Bergelson and Hindman's result to broader settings

Abstract

W. T. Gower generalized Hindman's Finite sum theorem over $X_{k}=\left\{ \left(n_{1},n_{2},\ldots,n_{k}\right):n_{1}\neq0\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_{i}}:i\in I\right\} $ and $\left\{ \mathbf{m_{i}}:i\in I\right\} $ such that the Gower subspace generated by $\left\{ \mathbf{n_{i}}:i\in I\right\} $ and set of all finite products of $\left\{ \mathbf{m_{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.