A necklace can be considered as a cyclic list of n red and n blue beads in an arbitrary order. In the necklace folding problem the goal is to find a large crossing-free matching of pairs of beads of different colors in such a way that there exists a “folding” of the necklace, that is a partition into two contiguous arcs, which splits the beads of any matching edge into different arcs.

We give counterexamples for some conjectures about the necklace folding problem, also known as the separated matching problem. The main conjecture (given independently by three sets of authors) states that $\mu =\frac{2}{3}$, where μ is the ratio of the maximum number of matched beads to the total number of beads.

We refute this conjecture by giving a construction which proves that $\mu \le 2-\sqrt{2}<0.5858\ll 0.66$. Our construction also applies to the homogeneous model when we match beads of the same color. Moreover, we also consider the problem where the two color classes do not necessarily have the same size.

一条项链可以被认为是任意顺序的n 个红色和n 个蓝色珠子的循环列表。在项链折叠问题中，目标是找到不同颜色的珠子对的大型无交叉匹配，这样项链就存在“折叠”，即分割成两个连续的弧线，将项链分开。任何匹配边缘的珠子变成不同的弧。

我们给出了一些关于项链折叠问题的猜想的反例，也称为分离匹配问题。主要猜想（由三组作者独立给出）指出$\mu =\frac{2}{3}$, 其中μ是匹配珠子的最大数量与珠子总数的比率。

我们通过给出一个证明来反驳这个猜想$\mu \le 2-\sqrt{2}<0.5858\ll 0.66$. 当我们匹配相同颜色的珠子时，我们的构造也适用于同质模型。此外，我们还考虑了两个颜色类别不一定具有相同大小的问题。