Ramsey properties of products of chains

Abstract

Let the product of t chains of equal size be called a t-dimensional grid. The main contribution of this paper is the following theorem: For every t-dimensional grid P there exists a (larger) t-dimensional grid Q such that if we color every comparability of Q with one of r colors, then there is a subposet P ′ of Q such that P ∼ = P ′ , and every comparability of P ′ is of the same color.We prove other related theorems for graphs, and a variation of our main tool: the Product Ramsey Theorem.