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

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• 作者：Ferdinand Ihringer ^{Ferdinand.Ihringer@math.uni-giessen.de" class="auth_mail" title="E-mail the corresponding author}
• 刊名：European Journal of Combinatorics
• 出版年：2016
• 出版时间：February 2016
• 年：2016
• 卷：52
• 期：part_PA
• 页码：27-39
• 全文大小：387 K

文摘

Let V$V$ be an n$n$-dimensional vector space over a finite field with q$q$ elements. Define a real-valued weight function on the 1$1$-dimensional subspaces of V$V$ such that the sum of all weights is zero. Let the weight of a subspace S$S$ be the sum of the weights of the 1$1$-dimensional subspaces contained in S$S$. In 1988 Manickam and Singhi conjectured that if n≥4k$n\ge 4k$, then the number of k$k$-dimensional subspaces with nonnegative weight is at least the number of k$k$-dimensional subspaces on a fixed 1$1$-dimensional subspace.

Recently, Chowdhury, Huang, Sarkis, Shahriari, and Sudakov proved the conjecture of Manickam and Singhi for n≥3k$n\ge 3k$. We modify the technique used by Chowdhury, Sarkis, and Shahriari to prove the conjecture for n≥2k$n\ge 2k$ if q$q$ is large. Furthermore, if equality holds and n≥2k+1$n\ge 2k+1$, then the set of k$k$-dimensional subspaces with nonnegative weight is the set of all k$k$-dimensional subspaces on a fixed 1$1$-dimensional subspace. With the exception of small q$q$, this result is the strongest possible, since the conjecture is no longer true for all n$n$ and k$k$ with k<n<2k$k<n<2k$.