Appendix: On Set Coverings in Cartesian Product Spaces

Abstract

Consider $(X,{\mathcal E})$, where X is a finite set and ${\mathcal E}$ is a system of subsets whose union equals X. For every natural number n ∈ℕ define the cartesian products X _n =∏₁ ⁿ X and ${\mathcal E}_n=\prod_1^n{\mathcal E}$. The following problem is investigated: how many sets of ${\mathcal E}_n$ are needed to cover X _n ? Let this number be denoted by c(n). It is proved that for all n ∈ℕ

$\exp\{C\cdot n\}\leq c(n)\leq\exp\{Cn+\log n+\log\log|X|\}+1.$

A formula for C is given. The result generalizes to the case where X and ${\mathcal E}$ are not necessarily finite and also to the case of non–identical factors in the product. As applications one obtains estimates on the minimal size of an externally stable set in cartesian product graphs and also estimates on the minimal number of cliques needed to cover such graphs.