More than twenty years ago, Manickam, Mikl\'{o}s, and Singhi conjectured that
for any integers n,k satisfying n≥4k, every set of n real numbers
with nonnegative sum has at least (k−1n−1)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≥eckloglogk. 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
(k−1n−1)+(k−1n−k−1)−1 subsets of cardinality k with
nonnegative sum.Comment: 15 pages, a section of Hilton-Milner type result adde