论文标题
几乎与家庭相交
Almost intersecting families
论文作者
论文摘要
令$ n> k> 1 $为整数，$ [n] = \ {1，\ ldots，n \} $。令$ \ Mathcal f $为$ k $ -subsets的家族〜$ [n] $。如果$ f \ cap f \ cap f'\ neq \ emptyset $用于所有$ f，f'\ in \ mathcal f $，则称为$ \ mathcal f $。如果它不是相交的，则几乎被称为相交，而是在\ mathcal f $中的每个$ f \ in \ Mathcal f $ in \ mathcal f $满足$ f \ cap f'= \ emptyset $中最多有一个$ f'\。 Gerbner等。证明，如果$ n \ geq 2k + 2 $，则$ | \ Mathcal f | \ leq {n -1 \选择k -1} $几乎相交的家庭。主要结果意味着相当强大，最可能的约束$ | \ Mathcal f | \ leq {n -1 \选择k -1} - {n -k -1 \选择k -1} + 2 $ for $ n>（2 + o（1））k $。
Let $n > k > 1$ be integers, $[n] = \{1, \ldots, n\}$. Let $\mathcal F$ be a family of $k$-subsets of~$[n]$. The family $\mathcal F$ is called intersecting if $F \cap F' \neq \emptyset$ for all $F, F' \in \mathcal F$. It is called almost intersecting if it is not intersecting but to every $F \in \mathcal F$ there is at most one $F'\in \mathcal F$ satisfying $F \cap F' = \emptyset$. Gerbner et al. proved that if $n \geq 2k + 2$ then $|\mathcal F| \leq {n - 1\choose k - 1}$ holds for almost intersecting families. The main result implies the considerably stronger and best possible bound $|\mathcal F| \leq {n - 1\choose k - 1} - {n - k - 1\choose k - 1} + 2$ for $n > (2 + o(1))k$.
