  In some recent papers the classical `splitting necklace theorem' is linked in
an interesting way with a geometric `pattern avoidance problem'. We explore the
topological constraints on the existence of a (relaxed) measurable coloring of
R^d such that any two distinct, non-degenerate cubes (parallelepipeds) are
measure discernible. For example, motivated by a conjecture of Lason, we show
that for every collection \mu_1,...,\mu_{2d-1} of 2d-1 continuous finite
measures on R^d, there exist two nontrivial axis-aligned d-dimensional cuboids
(rectangular parallelepipeds) C_1 and C_2 such that \mu_i(C_1)=\mu_i(C_2) for
each i=1,...,2d-1. We also show by examples that the bound 2d-1 cannot be
improved in general. These results are steps in the direction of studying
general topological obstructions for the existence of non-repetitive colorings
of measurable spaces.
