论文标题
最大的3个相交家族的最小尺寸：奇数
Maximal 3-wise Intersecting Families with Minimum Size: the Odd Case
论文作者
论文摘要
一个家庭$ \ MATHCAL {F} $在地面集合$ \ {1,2，\ ldots，n \} $是最大$ k $ - 如果每一个$ k $集合中的每一个集合中的$ \ Mathcal {f} $集合的每一个集合，则没有其他设置，并且无法添加其他设置，并且在$ Mathcal calcal cabal coplaince callcal calcal cabal cause propertrative propertrative。埃尔德（Erd）和克莱特曼（Kleitman）要求最大$ k $ wise的相交家庭的最小尺寸。补充了亨德里，隆德，汤普金斯和特兰的早期工作，他们以$ k = 3 $和大型$ n $回答了这个问题，我们以$ k = 3 $和大型奇数$ n $回答。我们表明，通过将地面套装分为两组$ a $和$ b $，尺寸几乎相等，并将所有适当的超级超级超级超级超级$ a $ a $ a $ a $和$ b $组成，从而获得了独特的最低家庭。我们证明的关键要素是埃利斯和苏达科夫关于所谓的$ 2 $生成器套装系统的稳定性结果。
A family $\mathcal{F}$ on ground set $\{1,2,\ldots, n\}$ is maximal $k$-wise intersecting if every collection of $k$ sets in $\mathcal{F}$ has non-empty intersection, and no other set can be added to $\mathcal{F}$ while maintaining this property. Erdős and Kleitman asked for the minimum size of a maximal $k$-wise intersecting family. Complementing earlier work of Hendrey, Lund, Tompkins and Tran, who answered this question for $k=3$ and large even $n$, we answer it for $k=3$ and large odd $n$. We show that the unique minimum family is obtained by partitioning the ground set into two sets $A$ and $B$ with almost equal sizes and taking the family consisting of all the proper supersets of $A$ and of $B$. A key ingredient of our proof is the stability result by Ellis and Sudakov about the so-called $2$-generator set systems.
