The Manickam–Miklós–Singhi conjectures for sets and vector spaces

Abstract

More than twenty-five years ago, Manickam, Miklós, and Singhi conjectured that for positive integers n, k with n ≥ 4k, every set of n real numbers with nonnegative sum has at least n−1 k−1 k-element subsets whose sum is also nonnegative. We verify this conjecture when n ≥ 8k 2 , which simultaneously improves and simplifies a bound of Alon, Huang, and Sudakov and also a bound of Pokrovskiy when k < 10 45 .Moreover, our arguments resolve the vector space analogue of this conjecture. Let V be an n-dimensional vector space over a finite field. Assign a real-valued weight to each 1-dimensional subspace in V so that the sum of all weights is zero. Define the weight of a subspace S ⊂ V to be the sum of the weights of all the 1-dimensional subspaces it contains. We prove that if n ≥ 3k, then the number of k-dimensional subspaces in V with nonnegative weight is at least the number of k-dimensional subspaces in V that contain a fixed 1-dimensional subspace. This result verifies a conjecture of Manickam and Singhi from 1988.

The Minimum Number of Nonnegative Edges in Hypergraphs

“…He reduced the conjecture to finding a k-uniform hypergraph on n vertices satisfying the MMS property (similar techniques were also employed earlier in [8]). The second conjecture, Conjecture 1.3, was very recently proved by Chowdhury, Sarkis, and Shahriari [3] simultaneously with our work. They also proved a quadratic bound n ≥ 8k 2 for sets.…”

Miklós–Manickam–Singhi conjectures on partial geometries

“…A 2012 breakthrough paper by Alon, Huang, and Sudakov [1] confirmed the conjecture for n ≥ 33k 2 . More recently, Chowdhury, Sarkis, and Shahriari [6] showed that the MMS conjecture is true for n ≥ 8k 2 . A linear bound was given by Pokrovskiy [21], who proved that the MMS conjecture holds, provided that n ≥ 10 47 k.…”

A note on the Manickam–Miklós–Singhi conjecture

“…Moreover, if equality holds, the family of k-element subsets with nonnegative sum is a star on x 1 , S ∈ X k : x 1 ∈ S . The proof of Theorem 1.2 is similar to that of Theorem 1.3 in [9], where we tackle the Manickam-Miklós-Singhi conjectures for sets and vector spaces simultaneously. For the reader's convenience, we present the calculations for the case of sets in full detail in this unpublished manuscript.…”

“…Although Conjecture 1.1 and the Erdős-Ko-Rado theorem share the same bound and extremal example, there is no obvious way to translate one question into the other. Conjecture 1.1 has attracted a lot of attention due to its connections with the Erdős-Ko-Rado theorem [1,2,3,4,5,6,7,8,9,11,13,14,16,17,18,19,20,21], but still remains open. For more than two decades, Conjecture 1.1 was known to hold only when k|n [18] or when n is at least an exponential function of k [4,6,17,21].…”