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

Abstract

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