A weight function from the set of all subsets of [n]={1,…,n} to the nonnegative real numbers is called shift-monotone in {m+1,…,n} if ω({a1,…,aj})⩾ω({b1,…,bj}) holds for all with , and if ω(A)⩾ω(B) holds for all A,B⊆[n] with A⊆B and B⧹A⊆{m+1,…,n}. A family is called intersecting in [m] if F∩G∩[m]≠∅ for all . Let . We show that is intersecting in is intersecting in [m]} provided that ω is shift-monotone in {m+1,…,n}. An application to the poset of colored subsets of a finite set is given.