论文标题
一些有限集的交点定理
Some intersection theorems for finite sets
论文作者
论文摘要
让$ n $，$ r $，$ k_1，\ ldots，k_r $和$ t $是带有$ r \ geq 2 $的正整数，以及$ \ m rathcal {f} _i \（1 \ leq i \ leq i \ leq r）$ $ k_i $ - $ k_i $ - $ n $ n $ -set $ -set $ -set $ -set $ -set $ -set $ -set $ -set $ -set $ -set $ -set $ -set $ -set $。 $ \ nathcal {f} _1，\ \ \ \ \ \ \ \ \ \ \ \ _2，\ ldots，\ Mathcal {f} _r $被称为$ r $ -r $ -cross $ t $ -t $ - t $ - ivecting如果$ | f_1 \ f_1 \ cap f_2 $ f_i \ in \ Mathcal {f} _i \（1 \ leq i \ leq r），$，如果$ | \ cap_ {1 \ cap_ {1 \ leq i \ leq i \ leq r} \ cap_ {f \ in \ nathcal {f} f} f} f | <t $。如果$ r $ -cross $ t $ -t $ diSterting家庭$ \ MATHCAL {f} _1，\ ldots，\ Mathcal {f} _r $ cablesy $ \ Mathcal {f} _1 = \ cdots = \ cdots = \ cdots = \ cdcal {f} $ r $ - 定型$ t $ - 更新家庭。在本文中，我们描述了非平凡的$ r $ - $ t $ t $ to的结构，这些家庭的规模最大，并为这些家庭带来稳定的结果。我们还确定了具有最大尺寸产物的非平凡$ 2 $ - 交叉$ t $ t $更新的家庭的结构。
Let $n$, $r$, $k_1,\ldots,k_r$ and $t$ be positive integers with $r\geq 2$, and $\mathcal{F}_i\ (1\leq i\leq r)$ a family of $k_i$-subsets of an $n$-set $V$. The families $\mathcal{F}_1,\ \mathcal{F}_2,\ldots,\mathcal{F}_r$ are said to be $r$-cross $t$-intersecting if $|F_1\cap F_2\cap\cdots\cap F_r|\geq t$ for all $F_i\in\mathcal{F}_i\ (1\leq i\leq r),$ and said to be non-trivial if $|\cap_{1\leq i\leq r}\cap_{F\in\mathcal{F}_i}F|<t$. If the $r$-cross $t$-intersecting families $\mathcal{F}_1,\ldots,\mathcal{F}_r$ satisfy $\mathcal{F}_1=\cdots=\mathcal{F}_r=\mathcal{F}$, then $\mathcal{F}$ is well known as $r$-wise $t$-intersecting family. In this paper, we describe the structure of non-trivial $r$-wise $t$-intersecting families with maximum size, and give a stability result for these families. We also determine the structure of non-trivial $2$-cross $t$-intersecting families with maximum product of their sizes.
