  A necklace can be considered as a cyclic list of $n$ red and $n$ blue beads
in an arbitrary order, and the goal is to fold it into two and find a large
cross-free matching of pairs of beads of different colors. We give a
counterexample for a conjecture about the necklace folding problem, also known
as the separated matching problem. The conjecture (given independently by three
sets of authors) states that $\mu=\frac{2}{3}$, where $\mu$ is the ratio of the
`covered' beads to the total number of beads.
  We refute this conjecture by giving a construction which proves that $\mu \le
2 \nolinebreak - \nolinebreak \sqrt 2 < 0.5858$. Our construction also applies
to the homogeneous model: when we are matching beads of the same color.
Moreover, we also consider the problem where the two color classes not
necessarily have the same size.
