Splitting multidimensional necklaces and measurable colorings of Euclidean spaces

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.