A note on the Manickam–Miklós–Singhi conjecture for vector spaces

Let VV be an nn-dimensional vector space over a finite field with qq elements. Define a real-valued weight function on the 11-dimensional subspaces of VV such that the sum of all weights is zero. Let the weight of a subspace SS be the sum of the weights of the 11-dimensional subspaces contained in SS. In 1988 Manickam and Singhi conjectured that if n≥4kn≥4k, then the number of kk-dimensional subspaces with nonnegative weight is at least the number of kk-dimensional subspaces on a fixed 11-dimensional subspace.Recently, Chowdhury, Huang, Sarkis, Shahriari, and Sudakov proved the conjecture of Manickam and Singhi for n≥3kn≥3k. We modify the technique used by Chowdhury, Sarkis, and Shahriari to prove the conjecture for n≥2kn≥2k if qq is large. Furthermore, if equality holds and n≥2k+1n≥2k+1, then the set of kk-dimensional subspaces with nonnegative weight is the set of all kk-dimensional subspaces on a fixed 11-dimensional subspace. With the exception of small qq, this result is the strongest possible, since the conjecture is no longer true for all nn and kk with k