两族Katona定理和Milner定理的类比
Analogues of Katona's and Milner's Theorems for two families
摘要：
设$n>s>0$为整数，$X$为$n$ -元集，$\mathscr{A}, \mathscr{B}\subset 2^X$为两族.如果所有$A\in\mathscr{A}, B\in \mathscr{B}$的$|A\cup B|\le s$，则$\mathscr{A}$和$\mathscr{B}$称为交叉$s$ -并集。假设$\mathscr{A}$和$\mathscr{B}$都不是空的，我们证明了几个最佳可能的界限。特别是，我们表明，$|\mathscr{A}|+|\mathscr{B}|\le 1+\sum\limits_{0\le i\le s}{{n}\choose{i}}$。假设$n\ge 2s$和$\mathscr{A},\mathscr{B}$是反链，我们证明了除非$\mathscr{A}=\{\emptyset\}$或$\mathscr{B}=\{\emptyset\}$，否则$|\mathscr{A}|+|\mathscr{B}|\le {{n}\choose{1}}+{{n}\choose{s-1}}$是反链。对三个族也建立了类似的结果。
英文摘要：
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.
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