Advances in the Minimum Overlap Problem Original Research Article

Given any positive integern, consider partitions of the integers 1, 2, …, 2ninto two disjoint classes {ai} and {bi} withnelements in each class. Denote byMkthe number of solutions ofk=ai−bj. In the minimum overlap problem, one seeks to estimateM(n)=min maxk Mkwhere the minimum is taken over all partitions. The best known results have been 0.3564less-than-or-equals, slantlim M(n)/nless-than-or-equals, slant0.4. The author improves the upper bound to lim M(n)/n<0.3857 by using the possibility of the truth of a conjecture. This conjecture is now a theorem of Swinnerton–Dyer, which allows the further improvement to 0.3820.