Multidimensional necklaces and measurable colorings of R^n
Combinatorics
2013-02-13 v3
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.
Keywords
Cite
@article{arxiv.1111.3519,
title = {Multidimensional necklaces and measurable colorings of R^n},
author = {Wojciech Lubawski},
journal= {arXiv preprint arXiv:1111.3519},
year = {2013}
}
Comments
newer version has beed uploaded: at arXiv:1209.1809