The three-permutations problem

Abstract Given any three permutations on {1,…, n }, we want to choose f :{1,…, n }→{-1, 1} so that the maximum absolute partial sum of f values over the permutations is minimized. The three-permutations problem is to determine the supremum of this minimum taken over all n and all triples of permutations on {1,…, n }. The only thing presently known about the supremum is that it is at least two. This paper establishes a result for a restricted case of the problem in which the maximum absolute partial sum for one of the three permutations equals 1. The supremum in this case is unbounded.