An Analogue of Gromov's Waist Theorem for Coloring the Cube
It is proved that if we partition a d-dimensional cube into small cubes and color the small cubes in colors then there exists a monochromatic connected component consisting of at least small cubes. Another proof of this result is given in Matdinov's preprint (Size of components of a cube coloring, arXiv:1111.3911, 2011).
graph coloring covering dimension waist inequality