论文标题
卡托纳和米尔纳定理的两个家庭的类似物
Analogues of Katona's and Milner's Theorems for two families
论文作者
论文摘要
令$ n> s> 0 $为整数，$ x $ a $ n $ element set和$ \ mathscr {a}，\ mathscr {b} \ subset 2^x $两个家庭。如果$ | a \ cup b | \ le s $用于\ inthscr {a}，b \ in \ mathscr {b} $，则$ \ mathscr {a} $和$ \ mathscr {b} $称为cross cross $ s-union。假设$ \ mathscr {a} $也不是$ \ mathscr {b} $是空的，我们证明了几个最佳范围。特别是，我们表明$ | \ Mathscr {a} |+| \ Mathscr {b} | \ le 1+ \ sum \ limits_ {0 \ le I \ le i \ le s} {{n} {n} \ select {n} \ select {i}}} $。假设$ n \ ge 2s $和$ \ Mathscr {a}，\ Mathscr {b} $是抗的，我们表明，$ | \ Mathscr {a} |+|+| \ Mathscr {b} | \ le le {n} $ \ mathscr {a} = \ {\ emptyset \} $或$ \ mathscr {b} = \ {\ emptyset \} $。也建立了三个家庭的类似结果。
Let $n>s>0$ be integers, $X$ an $n$-element set and $\mathscr{A}, \mathscr{B}\subset 2^X$ two families. If $|A\cup B|\le s$ for all $A\in\mathscr{A}, B\in \mathscr{B}$, then $\mathscr{A}$ and $\mathscr{B}$ are called cross $s$-union. Assuming that neither $\mathscr{A}$ nor $\mathscr{B}$ is empty, we prove several best possible bounds. In particular, we show that $|\mathscr{A}|+|\mathscr{B}|\le 1+\sum\limits_{0\le i\le s}{{n}\choose{i}}$. Supposing $n\ge 2s$ and $\mathscr{A},\mathscr{B}$ are antichains, we show that $|\mathscr{A}|+|\mathscr{B}|\le {{n}\choose{1}}+{{n}\choose{s-1}}$ unless $\mathscr{A}=\{\emptyset\}$ or $\mathscr{B}=\{\emptyset\}$. An analogous result for three families is established as well.
