On shifted intersecting families with respect to posets

In this paper, we show that for a shifted complex :Jr ~ 2 with respect to a poset P with minimum element 0 and an intersecting subfamily ~ ~ :Jr, #~ ~ #{FE:Jr;OEF}. We denote the set {1,2, .. ,n} by [n], the family of all subsets of a set X by 2x , #F denotes the number of elements of a set F. Let :Jr be a family of subsets of [n], i.e., :Jr = {F1, .. ,Fm } where F1 Fm are distinct subsets of [n]. A family g, is intersecting if for every F i , Fj E :Jr, Fi () Fj '* 0. For families ~, g, S;;; 2[n], ~ and :Jr are cross-intersecting if G () F '* 0 for V G E ~ and V F E :Jr. A family :Jr ~ 2[n] is called a complex if G S;;; F E g, implies G E :Jr. We already know the following results. For an intersecting family :Jr S;;; 2[n J, #:Jr ~ 2n1 ([1]) and for a complex :Jr S;;; 2[n] and cross-intersecting subfamilies ~, Je ~ :Jr, #~ + #Je ~ # g, ([4]). For F, G ~ [n], if there exists a one-to-one mapping I: F --> G with x ~ I(x) for each x E F, then we write F ~ G. :Jr ~ 2[n] Australasian Journal of Combinatorics ~ ( 1992 L pp. 53-58