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 says that t(k− 1) is an upper bound on the number of cutpoints of an opened t-coloured necklace so that the segments we get can be used to partition the set of vertices of the necklace into k subsets which have the property that every colour is represented by the same number of vertices in any element of the partition. The proof of our generalization of the BorsukUlam theorem uses a result from algebraic topology as a starting point and otherwise is purely combinatorial.