An improved bound for the Manickam–Miklós–Singhi conjecture

We show that for n > k 2 ( 4 e log k ) k , every set { x 1 , ⋯ , x n } of n real numbers with ∑ i = 1 n x i ≥ 0 has at least ( n − 1 k − 1 ) k -element subsets of a non-negative sum. This is a substantial improvement on the best previously known bound of n > ( k − 1 ) ( k k + k 2 ) + k , proved by Manickam and Miklós [9] in 1987.