An intersection theorem for weighted sets Original Research Article

A weight function ω : 2[n]→R⩾0 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 {a1,…,aj}, {b1,…,bj}⊆[n] with ai⩽bi, i=1,…,j, and if ω(A)⩾ω(B) holds for all A,B⊆[n] with A⊆B and B⧹A⊆{m+1,…,n}. A family F⊆2[n] is called intersecting in [m] if F∩G∩[m]≠∅ for all F,G∈F. Let ω(F)=∑F∈Fω(F). We show that max{ω(F): F⊆2[n], F is intersecting in [n]}=max{ω(F): F⊆2[n],F 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.