An intersection theorem for weighted sets

Abstract

A weight function $\text{ω}\text{:}\text{2}{}^{\mathrm{[n]}}\text{→}\text{R}{}_{\mathrm{⩾0}}$ 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,…,a_j})⩾ω({b₁,…,b_j}) holds for all $\text{{a}{}_1\text{,…,a}{}_{\mathrm{j}}\text{},}\text{{b}{}_1\text{,…,b}{}_{\mathrm{j}}\text{}⊆[n]}$ with $\text{a}{}_{\mathrm{i}}\text{⩽b}{}_{\mathrm{i}}\text{,}\text{i=1,…,j}$, and if ω(A)⩾ω(B) holds for all A,B⊆[n] with A⊆B and B⧹A⊆{m+1,…,n}. A family $\text{F}\text{⊆2}{}^{\mathrm{[n]}}$ is called intersecting in [m] if F∩G∩[m]≠∅ for all $\text{F,G∈}\text{F}$. Let $\text{ω(}\text{F}\text{)=∑}{}_{\mathrm{<mtext>F\in </mtext><mtext>F</mtext>}}\text{ω(F)}$. We show that $\text{max}\text{{ω(}\text{F}\text{)}\text{:}\text{F}\text{⊆2}{}^{\mathrm{[n]}}\text{,}\text{F}$ is intersecting in $\text{[n]}=}\text{max}\text{{ω(}\text{F}\text{)}\text{:}\text{F}\text{⊆2}{}^{\mathrm{[n]}}\text{,}\text{F}$ 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.