An analogue of Gromov's waist theorem for coloring the cube
  It is proved that if we partition a $d$-dimensional cube into $n^d$ small cubes and color the small cubes into $m+1$ colors then there exists a monochromatic connected component consisting of at least $f(d, m) n^{d-m}$ small cubes. 
Roman Karasev
math.CO math.AT