Discrepancy One among Homogeneous Arithmetic Progressions

We investigate a restriction of Paul Erdős’ well-known problem from 1936 on the discrepancy of homogeneous arithmetic progressions. We restrict our attention to a finite set S of homogeneous arithmetic progressions, and ask when the discrepancy with respect to this set is exactly 1. We answer this question when S has size four or less, and prove that the problem for general S is NP-hard, even for discrepancy 1.