On partitions of two-dimensional discrete boxes

Let A and B be finite sets and consider a partition of the discrete box A×B into sub-boxes of the form A^′×B^′ where A^′⊂A and B^′⊂B. We say that such a partition has the (k,ℓ)-piercing property for positive integers k and ℓ if for every a∈A the discrete line {a}×B intersects at least k sub-boxes and for every b∈B the line A×{b} intersects at least ℓ sub-boxes. We show that a partition of A×B that has the (k,ℓ)-piercing property must consist of at least (k−1)+(ℓ−1)+⌈2(k−1)(ℓ−1)⌉ sub-boxes. This bound is nearly tight (up to one additive unit) for all values of k and ℓ and is tight for infinitely many values of k and ℓ. As a corollary we get that the same bound holds for the minimum number of vertices of a graph whose edges can be colored red and blue such that every vertex is part of a red k-clique and a blue ℓ-clique.