The Manickam-Mikl\'os-Singhi Conjectures for Sets and Vector Spaces

More than twenty-five years ago, Manickam, Mikl'{o}s, and Singhi conjectured that for positive integers $n,k$ with $n\ge 4k$, every set of $n$ real numbers with nonnegative sum has at least $\left(\genfrac{}{}{0px}{}{n-1}{k-1}\right)$ $k$-element subsets whose sum is also nonnegative. We verify this conjecture when $n \geq 8k^{2}$, which simultaneously improves and simplifies a bound of Alon, Huang, and Sudakov and also a bound of Pokrovskiy when $k<{10}^{45}$. 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 $S\subset V$ to be the sum of the weights of all the $1$-dimensional subspaces it contains. We prove that if $n\ge 3k$, 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.