2014/05/05 by Ferdinand Ihringer, Ihringer, Ferdinand
Computer Science · Mathematics · #51E20 #52C10 #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Limits and Structures in Graph Theory
paper · pdf · doi:10.48550/arxiv.1405.0909
openalex publication_date 2014/05/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let V be an n-dimensional vector space over a finite field \mathbbFq. Define a real-valued weight function on the 1-dimensional vector spaces 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 et al. 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.