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 Erdos 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 $\mathbb{R}^{d}$ such that no cube has a fair splitting of size $t$, i.e., for each of the axes using at most $t$ hyperplanes orthogonal to it, provided $k\geq (t+4)^{d}-(t+3)^{d}+(t+2)^{d}-2^{d}+d(t+2)+3$. We also consider a discrete variant of the multidimensional necklace splitting problem in the spirit of the theorem of de Longueville and Živaljevic. The question of how many axes-aligned hyperplanes are needed for a fair splitting of a $d$-dimensional $k$-colored cube remains open.