Maximizing the number of nonnegative subsets

Abstract

Given a set of $n$ real numbers, if the sum of 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 $\left(\genfrac{}{}{0ex}{}{n-1}{k-1}\right)+\left(\genfrac{}{}{0ex}{}{n-1}{k-2}\right)+\cdots +\left(\genfrac{}{}{0ex}{}{n-1}{0}\right)+1$, 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.

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