Nonnegative k-sums, fractional covers, and probability of small deviations

Abstract

More than twenty years ago, Manickam, Miklós, and Singhi conjectured that for any integers n, k satisfying n ≥ 4k, every set of n real numbers with nonnegative sum has at least n−1 k−1 kelement subsets whose sum is also nonnegative. In this paper we discuss the connection of this problem with matchings and fractional covers of hypergraphs, and with the question of estimating the probability that the sum of nonnegative independent random variables exceeds its expectation by a given amount. Using these connections together with some probabilistic techniques, we verify the conjecture for n ≥ 33k 2 . This substantially improves the best previously known exponential lower bound n ≥ e ck log log k . In addition we prove a tight stability result showing that for every k and all sufficiently large n, every set of n reals with a nonnegative sum that does not contain a member whose sum with any other k − 1 members is nonnegative, contains at least n−1 k−1 + n−k−1 k−1 − 1 subsets of cardinality k with nonnegative sum.

A note on the Manickam–Miklós–Singhi conjecture

“…The main result of this paper verifies Conjecture 1.1 when n ≥ 8k 2 . In particular, Theorem 1.2 simultaneously improves and simplifies the bound n ≥ min{33k 2 , 2k 3 } of Alon, Huang, and Sudakov [2] and also the bound n ≥ 10 46 k of Pokrovskiy [20] when k < 10 45 . Note that there is no loss of generality in assuming that the n real numbers in Conjecture 1.1 are listed in decreasing order and sum to zero.…”

Fractional and integer matchings in uniform hypergraphs

“…The conjecture also has applications to the Manickam-Mikós-Singhi conjecture in number theory (for details see e.g. [2]). Despite its seeming simplicity Conjecture 1.1 is still wide open in general.…”

A linear bound on the Manickam–Miklós–Singhi conjecture

“…Tyomkyn [21] improved this bound to n ≥ k(4e log k) k ≈ e ck log log k . Recently Alon, Huang, and Sudakov [2] showed that the conjecture holds when n ≥ 33k 2 . Subsequently Frankl [8] gave an alternative proof of the conjecture in a range of the form n ≥ 3k 3 /2, and Chowdhury, Sarkis, and Shahriari [7] gave a proof of the conjecture in the range of the form n ≥ 8k 2 .…”