 
Summary: Nonnegative ksums, fractional covers, and probability of small
deviations
Noga Alon
Hao Huang
Benny Sudakov
Abstract
More than twenty years ago, Manickam, Mikl´os, and Singhi conjectured that for any integers
n, k satisfying n 4k, every set of n real numbers with nonnegative sum has at least n1
k1 k
element 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 33k2
. This substantially improves the best previously known exponential
lower bound n eck 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 n1
k1 +
