Upper bounds for the necklace folding problems
Endre Csóka, Zoltán L. Blázsik, Zoltán Király, Dániel Lenger  

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 µ = 2/3 , where µ is the ratio of the ‘covered’ beads to the total number of beads. We refute this conjecture by giving a construction which proves that µ ≤ 2 − √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 do not necessarily have the same size.