论文标题
关于非空的跨界家庭
On non-empty cross-intersecting families
论文作者
论文摘要
令$ 2^{[n]} $和$ \ binom {[n]} {i} $是$ \ {1,2，\ cdots，n \} $的所有$ i $ -subsets的电源集和类别。我们将两个家庭称为$ \ mathscr {a} $和$ \ Mathscr {b} $交叉间歇，如果$ a \ cap b \ neq \ neq \ emptySet $用于任何$ a \ in \ Mathscr {a} $和$ b \ in \ Mathscr {b} $。在本文中，我们表明，对于$ n \ geq k+l，l \ geq r \ geq 1，c> 0 $和$ \ mathscr {a} \ subseteq \ binom {[n]} {k} {k} {k}，\ mathscr {b}和$ \ Mathscr {b} $是交叉交流，$ \ binom {n-r} {l-r} \ leq | \ mathscr {b} | \ leq \ leq \ binom {n-1} {n-1} {l-1} {l-1} {l-1} {l-1} $ \ max \ left \ {\ binom {n} {k} - \ binom {n-r} {n-r} {k}+c \ binom {n-r} {n-r-r} {l-r}，\ \ \ \ \ \ \ binom {n-1} $ \ mathscr {a} $和$ \ Mathscr {b} $达到上限也是表征的。这概括了希尔顿和米尔纳的相应结果，$ c = 1 $和$ r = k = l $，这意味着Tokeushige和第二作者的结果（定理1.3）。
Let $2^{[n]}$ and $\binom{[n]}{i}$ be the power set and the class of all $i$-subsets of $\{1,2,\cdots,n\}$, respectively. We call two families $\mathscr{A}$ and $\mathscr{B}$ cross-intersecting if $A\cap B\neq \emptyset$ for any $A\in \mathscr{A}$ and $B\in \mathscr{B}$. In this paper we show that, for $n\geq k+l,l\geq r\geq 1,c>0$ and $\mathscr{A}\subseteq \binom{[n]}{k},\mathscr{B}\subseteq \binom{[n]}{l}$, if $\mathscr{A}$ and $\mathscr{B}$ are cross-intersecting and $\binom{n-r}{l-r}\leq|\mathscr{B}|\leq \binom{n-1}{l-1}$, then $$|\mathscr{A}|+c|\mathscr{B}|\leq \max\left\{\binom{n}{k}-\binom{n-r}{k}+c\binom{n-r}{l-r},\ \binom{n-1}{k-1}+c\binom{n-1}{l-1}\right\}$$ and the families $\mathscr{A}$ and $\mathscr{B}$ attaining the upper bound are also characterized. This generalizes the corresponding result of Hilton and Milner for $c=1$ and $r=k=l$, and implies a result of Tokushige and the second author (Theorem 1.3).
