论文标题
次优$ S $ UNION familes和$ s $ UNION抗矢量空间的敌
Suboptimal $s$-union familes and $s$-union antichains for vector spaces
论文作者
论文摘要
让$ v $是有限字段$ \ mathbb {f} _ {q} $上的$ n $维矢量空间，然后让$ \ mathcal {l}（v）= \ bigCup_ {0 \ leq k \ leq k \ leq n} \一个子空间$ \ MATHCAL {f} \ subseteq \ Mathcal {l}（v）$是$ s $ Union，如果DIM $（f+f'）\ leq s $保留所有$ f $，$ f $，$ f'\ in \ mathcal in \ mathcal {f} $。一个家庭$ \ MATHCAL {F} \ subseteq \ Mathcal {l}（v）$是抗小节，如果$ f \ nleq f'$持有任何两个不同的$ f，f'\ in \ mathcal {f} $。 Frankl和Tokushige在$ \ Mathcal {l}（v）$中以$ \ Mathcal {l}（v）$中的最佳$ s $ UNION家族取决于$ 2013 $。最近，弗兰克尔（Frankl）建立了$ \ mathcal {l}（v）$的$ s $ union $（s <n）$ andichains的红衣主教上限，而弗兰克尔（Frankl）最近建立了，而最佳的结构尚未显示。本文确定了所有用于矢量空间的次最佳$ s $ UNION家庭，然后调查$ S $ UNION敌金。对于$ s = n $或$ s = 2d <n $，我们完全确定所有最佳和次优$ s $ s $ sy-inion抗小节。对于$ s = 2d+1 <n $，我们证明了最佳抗小键要么是$ \ weft [v \ at d \ right] $，要么包含在$ \ weft [v \ at d \ at d \ right] \ bigcup \ bigcup \ weft [v \ at pop d+1 \ right] $，这使与阴影相关的平等满足。
Let $V$ be an $n$-dimensional vector space over the finite field $\mathbb{F}_{q}$, and let $\mathcal{L}(V)=\bigcup_{0\leq k\leq n}\left[V\atop k\right]$ be the set of all subspaces of $V$. A family of subspaces $\mathcal{F}\subseteq \mathcal{L}(V)$ is $s$-union if dim$(F+F')\leq s$ holds for all $F$, $F'\in\mathcal{F}$. A family $\mathcal{F}\subseteq \mathcal{L}(V)$ is an antichain if $F\nleq F'$ holds for any two distinct $F, F'\in \mathcal{F}$. The optimal $s$-union families in $\mathcal{L}(V)$ have been determined by Frankl and Tokushige in $2013$. The upper bound of cardinalities of $s$-union $(s<n)$ antichains in $\mathcal{L}(V)$ has been established by Frankl recently, while the structures of optimal ones have not been displayed. The present paper determines all suboptimal $s$-union families for vector spaces and then investigates $s$-union antichains. For $s=n$ or $s=2d<n$, we determine all optimal and suboptimal $s$-union antichains completely. For $s=2d+1<n$, we prove that an optimal antichain is either $\left[V\atop d\right]$ or contained in $\left[V\atop d\right]\bigcup \left[V\atop d+1\right]$ which satisfies an equality related with shadows.
