Necklace Bisection with One Cut Less than Needed - Simonyi, Gábor (The Electronic Journal of Combinatorics, 2008)
A well-known theorem of Goldberg and West states that two thieves can always split a necklace containing an even number of beads from each of k types fairly by at most k cuts. We prove that if we can use at most k-1 cuts and fair splitting is not possible then the thieves still have the following option. Whatever way they specify two disjoint sets D_1, D_2 of the types of beads with D_1D_2 eq, it will always be possible to cut the necklace (with k-1 cuts) so that the first thief gets more of those types of beads that are in D_1 and the second gets more of those in D_2, while the rest is divided equally. The proof combines the simple proof given by Alon and West to the original statement with a variant of the Borsuk-Ulam theorem due to Tucker and Bacon.