Maximizing the number of nonnegative subsets

TL;DR

This paper determines the maximum number of nonnegative-sum subsets in a set of real numbers under the condition that larger subsets have negative sums, providing two distinct proofs for the result.

Contribution

It solves a problem posed by Tsukerman by exactly characterizing the maximum count of nonnegative subsets under the given sum constraints.

Findings

Maximum number of nonnegative subsets is given by a specific binomial sum plus one.

Two different proofs are provided: one using a weighted Hall's Theorem, another extending Erd ext{"o}s-Ko-Rado Theorem.

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 $\binom{n-1}{k-1} + \binom{n-1}{k-2} + \cdots + \binom{n-1}{0}+1$, 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 Erd\H{o}s-Ko-Rado Theorem.