An analogue of Gromov's waist theorem for coloring the cube

Abstract

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.

References in corpus (3)

• A simpler proof of the Boros-Füredi-Bárány-Pach-Gromov theorem
• A topological central point theorem
• Size of components of a cube coloring

Cited by in corpus (4)

• Lower and upper bounds for the waists of different spaces
• Covering dimension using toric varieties
• Colouring Strong Products
• Size of components of a cube coloring