论文标题
交叉$ t $ - 对有限仿射空间的家庭
Cross $t$-intersecting families for finite affine spaces
论文作者
论文摘要
用$ \ Mathscr {m}（k，n）$表示$ ag（n，\ mathbb {f} _Q）$中的所有$ k $ -flats的集合。令$ \ mathscr {f} _1 \ subset \ subset \ mathscr {m}（k_1，n）$和$ \ mathscr {f} _2 \ subset \ subset \ subset \ mathscr {m mathscr {m}（k_2，k_2，n）$ $ f_1 \ in \ mathscr {f} _1 $和$ f_2 \ in \ mathscr {f} _2 $。我们说他们是交叉的$ t $ quistectrate家庭。此外，我们说，如果每个成员中的每个成员都包含$ ag（n，\ mathbb {f} _q）$中的固定$ t $ flats $，那么它们是微不足道的。在本文中，我们表明，具有最大尺寸产物的交叉$ T $更换家庭是微不足道的。我们还描述了具有最大尺寸产物的非平凡$ t $更换家庭的结构。
Denote the collection of all $k$-flats in $AG(n,\mathbb{F}_q)$ by $\mathscr{M}(k,n)$. Let $\mathscr{F}_1\subset\mathscr{M}(k_1,n)$ and $\mathscr{F}_2\subset\mathscr{M}(k_2,n)$ satisfy $\dim(F_1\cap F_2)\ge t$ for any $F_1\in\mathscr{F}_1$ and $F_2\in\mathscr{F}_2$. We say they are cross $t$-intersecting families. Moreover, we say they are trivial if each member of them contains a fixed $t$-flats in $AG(n,\mathbb{F}_q)$. In this paper, we show that cross $t$-intersecting families with maximum product of sizes are trivial. We also describe the structure of non-trivial $t$-intersecting families with maximum product of sizes.
