A New Quadratic Bound for the Manickam-Miklos-Singhi Conjecture

More than twenty-five years ago, Manickam, Miklos, and Singhi conjectured that for positive integers n,k with n ≥ 4k, every set of n real numbers with nonnegative sum has at least n−1 k−1 � k-element subsets whose sum is also nonnegative. We verify this conjecture when n ≥ 8k 2 , which simultaneously improves and simplifies a bound of Alon, Huang, and Sudakov and also a bound of Pokrovskiy when k < 10 45 .