The minimum number of nonnegative edges in hypergraphs

TL;DR

This paper proves a key property of hypergraphs related to nonnegative edges under vertex weightings, confirming a long-standing conjecture for large n and extending it to vector spaces, with implications in combinatorics.

Contribution

It establishes the MMS property for hypergraphs with equal codegrees when n > 10r^3, confirming the Manickam-Miklós-Singhi conjecture in this range and extending results to vector spaces.

Findings

Proves MMS property for hypergraphs with equal codegrees when n > 10r^3.

Verifies the MMS conjecture for n > 10k^3 in the context of real numbers.

Confirms the vector space MMS conjecture for n >= 4k over finite fields.

Abstract

An r-unform n-vertex hypergraph H is said to have the Manickam-Mikl\'os-Singhi (MMS) property if for every assignment of weights to its vertices with nonnegative sum, the number of edges whose total weight is nonnegative is at least the minimum degree of H. In this paper we show that for n>10r^3, every r-uniform n-vertex hypergraph with equal codegrees has the MMS property, and the bound on n is essentially tight up to a constant factor. This result has two immediate corollaries. First it shows that every set of n>10k^3 real numbers with nonnegative sum has at least $\binom{n-1}{k-1}$ nonnegative k-sums, verifying the Manickam-Mikl\'os-Singhi conjecture for this range. More importantly, it implies the vector space Manickam-Mikl\'os-Singhi conjecture which states that for n >= 4k and any weighting on the 1-dimensional subspaces of F_q^n with nonnegative sum, the number of nonnegative k-dimensional subspaces is at least ${n-1 \brack k-1}_q$. We also discuss two additional generalizations, which can be regarded as analogues of the Erd\H{o}s-Ko-Rado theorem on k-intersecting families.