On 2-colored graphs and partitions of boxes

Abstract

We prove that if the edges of a graph G can be colored blue or red in such a way that every vertex belongs to a monochromatic k-clique of each color, then G has at least 4(k−1) vertices. This confirms a conjecture of Bucic et al. (0000), and thereby solves the 2-dimensional case of their problem about partitions of discrete boxes with the k-piercing property. We also characterize the case of equality in our result.