SPLITTING NECKLACES AND A GENERALIZATION OF THE BORSUK-ULAM ANTIPODAL THEOREM
Jerzy Wojciechowski · 1996
Abstract. 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 Borsuk-Ulam theorem uses a result from algebraic topology as a starting point and otherwise is purely combinatorial. 1 1.