Size of components of a cube coloring

TL;DR

This paper establishes a lower bound on the size of the largest color class in a d-dimensional lattice cube coloring, given constraints on face colorings, revealing a fundamental combinatorial property.

Contribution

It introduces a new bound relating the number of colors on faces to the minimal size of a color class in high-dimensional lattice colorings.

Findings

One color appears at least f(d,m)*n^(d-m) times.

The bound depends on the dimension d and parameter m.

Provides a combinatorial limit for face coloring configurations.

Abstract

Suppose a d-dimensional lattice cube of size n^d is colored in several colors so that no face of its triangulation (subdivision of the standard partition into n^d small cubes) is colored in m+2 colors. Then one color is used at least f(d,m)*n^(d-m) times.