Multidimensional necklaces and measurable colorings of Rn
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 - 2d + d(t+2) +3 different colors) of Rn 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 Rn such that no two disjoint n-dimensional cubes have the same measure of each color.
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