A set coverage problem

This paper shows that with B = {1, 2, . . . , n}, the smallest k such that (B ×B)− {(j, j) | j ∈ B} = k ⋃ i=1 (Ci ×Di) is s(n), where s(n) is the smallest integer k such that n 6 ( k b k 2 c ) . This provides a simple set-based formulation and a new proof of a result for boolean ranks [2] and biclique covering of bipartite graphs [1, 5], making these intricate results more accessible.