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 $n > (2 + o(1))k$.
翻译：$ > k > 1美元 是整数, $[ $] = = = {1,\ ldots, n} $。 $mathcal F$ 是一个以美元为单位的子集- $[ n] 美元。 如果所有美元都是F$\ cap F'\\ neq + 2美元, F'\\ leq = mathcal F$, F' = $, f' = = $ = = 1, 则家庭F$ = 1, = = = = $1, = f = = 美元= 美元= 美元= 美元。 Gerbner et al. 证明, 如果美元\ geq 2 k + 2 美元, $ * * * * * leq {n - 1\ chose k- 1}, 它几乎是交叉的。 其主要结果意味着对于几乎是相当强大和最好的约束 $ *\\ k\ k\\\ k\ k\ k\ k\ k\\ k\ k\ k\ k\ k\ k\ k\ k\ k\ k\ k\ k\ k\ k\ k\ k\ k\ k\ k\ k\ k\ k\ k\ k\ k\ k\ k\ = 1美元。
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