An intersection theorem for weighted sets

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({a_l,...,a_j})≥ω({b_l,...,b _j}) holds for all {a_l,...,a_j}, {b_l,...,b_j} ⊆ [n] with a_i≤b_i, 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) = Σ_F∈ℱ ^ω(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 ⊆ is shift-monotone in {m + 1,...,n}. An application to the poset of colored subsets of a finite set is given.