Obstacles for splitting multidimensional necklaces

Abstract

The well-known “necklace splitting theorem” of Alon (1987) asserts that every k k -colored necklace can be fairly split into q q parts using at most t t cuts, provided k ( q − 1 ) ≤ t k(q-1)\leq t . In a joint paper with Alon et al. (2009) we studied a kind of opposite question. Namely, for which values of k k and t t is there a measurable k k -coloring of the real line such that no interval has a fair splitting into 2 2 parts with at most t t cuts? We proved that k > t + 2 k>t+2 is a sufficient condition (while k > t k>t is necessary). We generalize this result to Euclidean spaces of arbitrary dimension d d , and to arbitrary number of parts q q . We prove that if k ( q − 1 ) > t + d + q − 1 k(q-1)>t+d+q-1 , then there is a measurable k k -coloring of R d \mathbb {R}^d such that no axis-aligned cube has a fair q q -splitting using at most t t axis-aligned hyperplane cuts. Our bound is of the same order as a necessary condition k ( q − 1 ) > t k(q-1)>t implied by Alon’s 1987 work. Moreover, for d = 1 , q = 2 d=1,q=2 we get exactly the result of the 2009 work. Additionally, we prove that if a stronger inequality k ( q − 1 ) > d t + d + q − 1 k(q-1)>dt+d+q-1 is satisfied, then there is a measurable k k -coloring of R d \mathbb {R}^d with no axis-aligned cube having a fair q q -splitting using at most t t arbitrary hyperplane cuts. The proofs are based on the topological Baire category theorem and use algebraic independence over suitably chosen fields.