The Minimum Number of Nonnegative Edges in Hypergraphs

Hao Huang,Benny Sudakov

doi:10.37236/4402

Abstract

An $r$-uniform $n$-vertex hypergraph $H$ is said to have the Manickam-Miklós-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-Miklós-Singhi conjecture for this range. More importantly, it implies the vector space Manickam-Miklós-Singhi conjecture which states that for $n \ge 4k$ and any weighting on the $1$-dimensional subspaces of $\mathbb{F}_{q}^n$ 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 Erdős-Ko-Rado theorem on $k$-intersecting families.

Highlights

Given an r-uniform n-vertex hypergraph H with minimum degree δ(H), suppose every vertex has a weight wi such that w1 + · · · + wn 0
We show that for every r-uniform n-vertex hypergraph with equal codegrees and n > 10r3, the minimum number of nonnegative edges is always achieved by the stars
A r–(n, t, λ) block design is a collection of t-subsets of [n] such that every r elements are contained in exactly λ subsets

Summary

Introduction

Given an r-uniform n-vertex hypergraph H with minimum degree δ(H), suppose every vertex has a weight wi such that w1 + · · · + wn 0. A hypergraph H with minimum degree δ(H) has the MMS property if for every weighting w : V (H) → R satisfying x∈v(H) w(x) 0, the number of nonnegative edges is at least δ(H). A linear bound n 1046k was obtained by Pokrovskiy [12] He reduced the conjecture to finding a k-uniform hypergraph on n vertices satisfying the MMS property (similar techniques were employed earlier in [9]). The following theorem says that for large n, this hypergraph has the MMS property Note that this is not implied by Theorem 1.4, because the codegree of two vertices (as k-subsets) depends on the size of their intersection. The final section contains some open problems and further research directions

Equal codegrees and MMS property

Two additional generalizations

Generalization of MMS

Generalization of vector MMS

Concluding Remarks