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

TL;DR

This paper extends Gromov's waist theorem to colored cubes, proving that in any coloring of a subdivided cube, a large monochromatic connected component must exist, with size depending on the dimension and number of colors.

Contribution

It introduces a novel analogue of Gromov's waist theorem specifically for cube colorings, establishing lower bounds on monochromatic component sizes.

Findings

Existence of large monochromatic connected components in colored subdivided cubes

Quantitative bounds depending on dimension and number of colors

Generalization of geometric measure theory results to discrete cube colorings

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.