  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.

Combinatorics;General Topology;Metric Geometry