Maximizing the Number of Nonnegative Subsets

Given a set of n real numbers, if the sum of the elements of every subset of size larger than k is negative, what is the maximum number of subsets of nonnegative sum? In this note we show that the answer is (n−1 k−1 ) + (n−1 k−2 ) + · · ·+ (n−1 0 ) + 1, settling a problem of Tsukerman. We provide two proofs; the first establishes and applies a weighted version of Hall’s theorem, and the second is based on an extension of the nonuniform Erdős–Ko–Rado theorem.