The three-permutations problem (Q908913)

Given a set \(N=\{1,2,3,...,n\}\) and any three permutations on it, it is desired to choose f: \(N\to \{-1,1\}\) in such a way so as to minimize the maximum of absolute partial sum of f values. The three-permutation problem is to obtain the supremum of this minimum. Nothing is known about this supremum except that it is greater than or equal to 2. In this paper the authors consider a special case of this problem in which the maximum absolute partial sum for one of the three permutations is 1. It is shown that for this special case the supremum is unbounded.