论文标题
一组亚集的跨交叉族家庭的最大尺寸总和
The maximum sum of sizes of cross-intersecting families of subsets of a set
论文作者
论文摘要
一组被称为家庭。如果$ \ Mathcal {a} $相交的每个成员相交的每个成员$ \ Mathcal {b} $，则两个家庭$ \ MATHCAL {a} $和$ \ MATHCAL {B} $ sets $ sets $。对于任何两个整数$ n $和$ k $，带有$ 1 \ leq k \ leq n $，让$ {[n] \ select \ leq k} $表示$ [n] = \ {1，\ dots，n \} $的子集的家族。我们表明，如果$ {a} $是$ {[n] \ select \ leq r} $的非空的亚家族，$ \ nathcal {b} $是$ {[n] \ select of $ {[n] $ \ MATHCAL {B} $是交叉截距，然后\ [| \ Mathcal {a} | + | \ Mathcal {B} | \ leq 1 + \ sum_ {i = 1}^s \ left（{n \ select i} - {n -r \ select i} \ right），\]，\]，如果$ \ mathcal {a} = \ {a} = \ {a} = \ {[r]那相交$ [r] $。
A set of sets is called a family. Two families $\mathcal{A}$ and $\mathcal{B}$ of sets are said to be cross-intersecting if each member of $\mathcal{A}$ intersects each member of $\mathcal{B}$. For any two integers $n$ and $k$ with $1 \leq k \leq n$, let ${[n] \choose \leq k}$ denote the family of subsets of $[n] = \{1, \dots, n\}$ that have at most $k$ elements. We show that if $\mathcal{A}$ is a non-empty subfamily of ${[n] \choose \leq r}$, $\mathcal{B}$ is a non-empty subfamily of ${[n] \choose \leq s}$, $r \leq s$, and $\mathcal{A}$ and $\mathcal{B}$ are cross-intersecting, then \[|\mathcal{A}| + |\mathcal{B}| \leq 1 + \sum_{i=1}^s \left({n \choose i} - {n-r \choose i} \right),\] and equality holds if $\mathcal{A} = \{[r]\}$ and $\mathcal{B}$ is the family of sets in ${[n] \choose \leq s}$ that intersect $[r]$.
