A counterexample to Beck's conjecture on the discrepancy of three permutations

Abstract

Given three permutations on the integers 1 through n, consider the set system consisting of each interval in each of the three permutations. Jozsef Beck conjectured (c. 1987) that the discrepancy of this set system is O(1). We give a counterexample to this conjecture: for any positive integer n = 3^k, we exhibit three permutations whose corresponding set system has discrepancy Omega(log(n)). Our counterexample is based on a simple recursive construction, and our proof of the discrepancy lower bound is by induction. This example also disproves a generalization of Beck's conjecture due to Spencer, Srinivasan and Tetali, who conjectured that a set system corresponding to l permutations has discrepancy O(sqrt(l)).