  A (continuous) necklace is simply an interval of the real line colored
measurably with some number of colors. A well-known application of the
Borsuk-Ulam theorem asserts that every $k$-colored necklace can be fairly split
by at most $k$ cuts (from the resulting pieces one can form two collections,
each capturing the same measure of every color). Here we prove that for every
$k\geq 1$ there is a measurable $(k+3)$-coloring of the real line such that no
interval can be fairly split using at most $k$ cuts. In particular, there is a
measurable $4$-coloring of the real line in which no two adjacent intervals
have the same measure of every color. An analogous problem for the integers was
posed by Erd\H{o}s in 1961 and solved in the affirmative in 1991 by Ker\"anen.
Curiously, in the discrete case the desired coloring also uses four colors.
