论文标题
关于交叉斑点家庭
On Cross-intersecting Sperner Families
论文作者
论文摘要
如果$ x \ cap y \ neq \ neq \ neq \ neq \ neq \ neq \ emberySet $ for hasts $ x \ in \ mathscr {a a} $，则两个集合$ \ mathscr {a} $和$ \ mathscr {b} $，如果$ x \ cap y \ neq \ nekyset $，则$ \ mathscr {b} $被认为是交叉交流的。给定了两个交叉切断的斑点家庭（或抗小族）$ \ mathscr {a} $和$ \ mathscr {b} $ of $ \ mathbb {n} _n $，我们证明了$ | \ m m mathscr {a} | 2 {{n} \选择{\ lceil {n/2} \ rceil}} $如果$ n $是奇怪的，而$ | \ mathscr {a} |+|+| \ mathscr {b} | \ le le {n} {n} {n} $ n $偶数。此外，确定了$ \ mathscr {a} $和$ \ Mathscr {B} $的所有极端和几乎超级家庭。
Two sets $\mathscr{A}$ and $\mathscr{B}$ are said to be cross-intersecting if $X\cap Y\neq\emptyset$ for all $X\in\mathscr{A}$ and $Y\in\mathscr{B}$. Given two cross-intersecting Sperner families (or antichains) $\mathscr{A}$ and $\mathscr{B}$ of $\mathbb{N}_n$, we prove that $|\mathscr{A}|+|\mathscr{B}|\le 2{{n}\choose{\lceil{n/2}\rceil}}$ if $n$ is odd, and $|\mathscr{A}|+|\mathscr{B}|\le {{n}\choose{n/2}}+{{n}\choose{(n/2)+1}}$ if $n$ is even. Furthermore, all extremal and almost-extremal families for $\mathscr{A}$ and $\mathscr{B}$ are determined.
