How different can two intersecting families be?

Abstract

To measure the difference between two intersecting families F, G ⊆ 2[n] we introduce the quantity D(F,G) = |{(F,G) : F ∈ F, G ∈ G, F ∩ G = θ}|We prove that if F is k-uniform and G is l-uniform, then for large enough n and for any i ≠ j Fi = {F ⊆ [n] : i ∈ F, |F| = k} and Fi = {F ⊆[n] : j ∈ F, |F| = l} form an optimal pair of families (that is D(F, G} ≤ D(F i, Fj) for all uniform and intersecting F and G), while in the non-uniform case any pair of the form Fi = {F ⊆ [n] : i ∈ F} and Fj = {F ⊆ [n] : j ∈F} is optimal for any n.