The Manickam–Miklós–Singhi conjectures for sets and vector spaces

More than twenty-five years ago, Manickam, Miklós, and Singhi conjectured that for positive integers n,kn,k with n≥4kn≥4k, every set of n   real numbers with nonnegative sum has at least (n−1k−1)k  -element subsets whose sum is also nonnegative. We verify this conjecture when n≥8k2n≥8k2, which simultaneously improves and simplifies a bound of Alon, Huang, and Sudakov and also a bound of Pokrovskiy when k<1045k<1045.Moreover, our arguments resolve the vector space analogue of this conjecture. Let V be an n-dimensional vector space over a finite field. Assign a real-valued weight to each 1-dimensional subspace in V   so that the sum of all weights is zero. Define the weight of a subspace S⊂VS⊂V to be the sum of the weights of all the 1-dimensional subspaces it contains. We prove that if n≥3kn≥3k, then the number of k-dimensional subspaces in V with nonnegative weight is at least the number of k-dimensional subspaces in V that contain a fixed 1-dimensional subspace. This result verifies a conjecture of Manickam and Singhi from 1988.