A fixed box theorem for the cartesian product of graphs and metric spaces Original Research Article

We prove the following result: Let (X, δ) = ∏i∈I(Xi, δi), be a product of a finite number of finite metric spaces, where the distance in X is the sum of the coordinate distances. Then for each non-expanding map ϕ : X → X, there exists a set R ⊆ X such that R coincides with the product of its projections and ϕ(R) = R. This extends known results on cubes and Hamming graphs.