On covering a product of sets with products of their subsets

Suppose that S₁,…,S_N are collections of subsets of X₁,…,X_N, respectively, such that n_i subsets belonging to S_i, and no fewer, cover X_i for all i. the main result of this paper is that to cover X₁ x…x X_N requires no fewer than σ^N_i=1 (n_i–1) + 1 and no more than Π^N_i=1n_i subsets of the form A₁ x…x A_N, where A_i∈S₁foralli. Moreo ver, these bounds cannot be improved. Identical bounds for the spanning number of a normal product of graphs are also obtained.