子集格的一个特殊子集的交族

Intersecting Families in a Subset of Boolean Lattices

设正整数n,r,l,s满足r<l<s≤n/2,X1,X2,X3是两两不交的n元集合,定义Ω={X A∈(:{A∩X1,A∩X2,A∩X3}={r,l,s}r+l+s)},其中X=X1∪X2∪X3.在本文中,我们将证明,若F是Ω的一个交族,则{(n-1F≤2r-1)(nl)(ns)(n-1+l-1)(nr)(ns)(n-1+s-1)(nr)(nl)},且等号成立当且仅当F={A∈F:a∈A},a∈X.

Let n,r,l and s be positive integers with r l s≤n /2,and let X1,X2 and X3be three pairwise disjoint sets with the same sizen. Set Ω = A∈Xr ＋ l ＋s： A∩X1,A∩X2{,A∩X }3= {r,l,}{ }s,where X = X1∪X2∪X3. In this paper,we prove that if F is an intersecting family of Ω,then F ≤2n- 1r- 1nlns＋n- 1l- 1nrns＋n- 1s- 1nrn l,and equality holds if and only if F = {A∈F： a∈A},for some a∈X.