Multidimensional necklaces and measurable colorings of R^n

TL;DR

This paper constructs finite colorings of R^n that prevent fair splitting of cubes with limited cuts, extending the understanding of measurable colorings and their combinatorial properties.

Contribution

It proves the existence of finite colorings of R^n that prohibit fair splitting of cubes with a bounded number of cuts, generalizing previous splitting theorems.

Findings

Existence of finite colorings preventing fair splits with t cuts

Colorings with at least (t+4)^d - (t+3)^d + (t+2)^d - 2^d + d(t+2) +3 colors

No two disjoint cubes have identical color measures

Abstract

A well known generalization of Alon's "splitting nacklace theorem" by Longueville and Zivaljevic states that every k-colored n-dimensional cube can be fairly split using only k cuts in each dimension. Here we prove that for every t there exist a finite coloring (with at least (t+4)^d - (t+3)^d + (t+2)^d - 2^d + d(t+2) +3 different colors) of R^n such that no n-dimensional cube can be fairly split using at most t cuts in each dimension. In particular there is a finite coloring of R^n such that no two disjoint n-dimensional cubes have the same measure of each color.