  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, Lidicky,
Long, and Wagner (arXiv:1805.11278[math.CO]) 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.
