Splitting multidimensional necklaces and measurable colorings of Euclidean spaces

TL;DR

This paper extends the multidimensional necklace splitting theorem to higher-dimensional Euclidean spaces, providing bounds on colorings that prevent fair splits with limited hyperplanes, and explores related discrete problems.

Contribution

It generalizes the necklace splitting theorem to higher dimensions and establishes bounds on colorings that avoid fair splits with a limited number of hyperplanes.

Findings

Existence of k-colorings in R^d avoiding fair t-splits with hyperplanes

Derived bounds on the number of colors needed for such colorings

Open problem on the number of hyperplanes for discrete cube splits

Abstract

A necklace splitting theorem of Goldberg and West asserts that any k-colored (continuous) necklace can be fairly split using at most k cuts. Motivated by the problem of Erd\H{o}s on strongly nonrepetitive sequences, Alon et al. proved that there is a (t+3)-coloring of the real line in which no necklace has a fair splitting using at most t cuts. We generalize this result for higher dimensional spaces. More specifically, we prove that there is k-coloring of R^{d} such that no cube has a fair splitting of size t (using at most t hyperplanes orthogonal to each of the axes), provided k>(t+4)^{d}-(t+3)^{d}+(t+2)^{d}-2^{d}+d(t+2)+3. We also consider a discrete variant of the multidimensional necklace splitting problem in the spirit of the theorem of de Longueville and \v{Z}ivaljevi\'c. The question how many axes aligned hyperplanes are needed for a fair splitting of a d-dimensional k-colored cube remains open.