A note on the Manickam–Miklós–Singhi conjecture for vector spaces. (February 2016) - Ihringer, Ferdinand
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 ≥ 4 k, 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 ≥ 3 k . We modify the technique used by Chowdhury, Sarkis, and Shahriari to prove the conjecture for n ≥ 2 k if q is large. Furthermore, if equality holds and n ≥ 2 k + 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 < 2 k .