Maximizing the Number of Nonnegative Subsets

Abstract

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 $\binom{n-1}{k-1} + \binom{n-1}{k-2} + \cdots + \binom{n-1}{0}+1$, 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 Erdös--Ko--Rado theorem.