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- W2949169857 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." @default.
- W2949169857 created "2019-06-27" @default.
- W2949169857 creator A5063469618 @default.
- W2949169857 date "2013-04-19" @default.
- W2949169857 modified "2023-09-27" @default.
- W2949169857 title "Obstacles for splitting multidimensional necklaces" @default.
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- W2949169857 doi "https://doi.org/10.1090/proc/12611" @default.
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