Formally, the discrepancy of an interval is defined as the difference between the number of white elements and the number of black elements in that interval; the objective is to color the elements such that the maximum discrepancy of an interval in each of the permutations is as small as possible.
The discrepancy of the permutations p1, ...,pm is the minimum, over all black-white colorings of the integers in [n], of the maximum over all intervals, of the difference between the number of white and black integers in the interval.
Jiang, Kulkarni and Singla call the setting with one permutation Online Interval Discrepancy.
They prove that:[1]: Sec.3.2 The proof extends for the case of two permutations, which they call Online Stripe Discrepancy.
Results in discrepancy of permutations have been used in the computation of Agreeable subsets, as well as in Online fair division.