An improved bound for the Manickam-Miklós-Singhi conjecture

We show that for n>k2(4elogk)k, every set {x1,⋯,xn} of n real numbers with ∑i=1nxi≥0 has at least (n−1k−1)k-element subsets of a non-negative sum. This is a substantial improvement on the best previously known bound of n>(k−1)(kk+k2)+k, proved by Manickam and Miklós [9] in 1987.