An intersection theorem for weighted sets

A weight function ! : 2 → 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 !({a1; : : : ; aj})¿!({b1; : : : ; bj}) holds for all {a1; : : : ; aj}; {b1; : : : ; bj}⊆ [n] with ai6bi; i = 1; : : : ; j, and if !(A)¿!(B) holds for all A; B⊆ [n] with A⊆B and B\A⊆{m + 1; : : : ; n}. A family F⊆ 2 is called intersecting in [m] if F ∩ G ∩ [m] = ∅ for all F; G ∈ F. Let !(F) = ∑F∈F !(F). We show that max{!(F): F⊆ 2; F is intersecting in [n]} = max{!(F): F⊆ 2;F is intersecting in [m]} provided that ! is shift-monotone in {m+1; : : : ; n}. An application to the poset of colored subsets of a 0nite set is given. c © 2001 Elsevier Science B.V. All rights reserved.