Obstacles for splitting multidimensional necklaces

TL;DR

This paper explores the limitations of fair splitting of colored Euclidean spaces, establishing bounds on the number of colors and cuts needed to prevent fair divisions of axis-aligned cubes.

Contribution

It generalizes previous results to higher dimensions and multiple parts, providing new bounds on colorings that obstruct fair splittings with limited cuts.

Findings

For $k(q-1)>t+d+q-1$, there exists a coloring with no fair $q$-splitting of cubes.

The bounds are tight up to a constant factor, matching necessary conditions.

The methods involve topological Baire category theorem and algebraic independence.

Abstract

The well-known "necklace splitting theorem" of Alon asserts that every $k$-colored necklace can be fairly split into $q$ parts using at most $t$ cuts, provided $k(q-1)\leq t$. In a joint paper with Alon et al. we studied a kind of opposite question. Namely, for which values of $k$ and $t$ there is a measurable $k$-coloring of the real line such that no interval has a fair splitting into $2$ parts with at most $t$ cuts? We proved that $k>t+2$ is a sufficient condition (while $k>t$ is necessary). We generalize this result to Euclidean spaces of arbitrary dimension $d$, and to arbitrary number of parts $q$. We prove that if $k(q-1)>t+d+q-1$, then there is a measurable $k$-coloring of $\mathbb{R}^d$ such that no axis-aligned cube has a fair $q$-splitting using at most $t$ axis-aligned hyperplane cuts. Our bound is of the same order as a necessary condition $k(q-1)>t$ implied by a theorem of Alon. Moreover for $d=1,q=2$ we get exactly the result of of Alon et al. Additionally, we prove that if a stronger inequality $k(q-1)>dt+d+q-1$ is satisfied, then there is a measurable $k$-coloring of $\mathbb{R}^d$ with no axis-aligned cube having a fair $q$-splitting using at most $t$ arbitrary hyperplane cuts. The proofs are based on the topological Baire category theorem and use algebraic independence over suitably chosen fields.