Splitting Necklaces and a Generalization of the Borsuk-Ulam Antipodal Theorem

We prove a very natural generalization of the Borsuk-Ulam antipodal theorem and deduce from it, in a very straightforward way, the celebrated result of Alon [1] on splitting necklaces. Alon’s result states that \(t(k-1)\) is an upper bound on the number of cutpoints of an opened \(t\)-colored necklace such that the segments obtained can be used to partition the set of vertices of the necklace into $k$ subsets with the property that every color is represented by the same number of vertices in any element of the partition. The proof of our generalization of the Borsuk-Ulam theorem uses a result from algebraic topology as a starting point and is otherwise purely combinatorial.