MMS-type problems for Johnson scheme

In the current work we consider the minimization problems for the number of nonzero or negative values of vectors from the first and second eigenspaces of the Johnson scheme respectively. We obtain several new lower and upper bounds for these characteristics based on equitable partitions and properties of eigenspaces of the Johnson scheme. The topic is a meeting point for generalizations of the Manikam–Miklós–Singhi conjecture and the minimum support problem for the eigenspaces of the Johnson graph, asymptotically solved by authors in a recent paper.