Ramsey properties of products of chains

Abstract

Let $\mathbf{k}$ denote the totally ordered set (or chain) on $k$ elements. The product ${\mathbf{k}}^t=\mathbf{k}\times \cdots \times \mathbf{k}$ is a poset called a grid. This paper discusses several loosely related results on the Ramsey theory of grids. Most of the results involve some application of the Product Ramsey Theorem.