Cross-intersecting families and primitivity of symmetric systems

Let X be a finite set and p ⊆ 2 X , the power set of X, satisfying three conditions: (a) p is an ideal in 2 X , that is, if A ∈ p and B ⊂ A , then B ∈ p ; (b) for A ∈ 2 X with | A | ⩾ 2 , A ∈ p if { x , y } ∈ p for any x , y ∈ A with x ≠ y ; (c) { x } ∈ p for every x ∈ X . The pair ( X , p ) is called a symmetric system if there is a group Γ transitively acting on X and preserving the ideal p . A family { A 1 , A 2 , … , A m } ⊆ 2 X is said to be a cross- p -family of X if { a , b } ∈ p for any a ∈ A i and b ∈ A j with i ≠ j . We prove that if ( X , p ) is a symmetric system and { A 1 , A 2 , … , A m } ⊆ 2 X is a cross- p -family of X, then ∑ i = 1 m | A i | ⩽ { | X | if m ⩽ | X | α ( X , p ) , m α ( X , p ) if m ⩾ | X | α ( X , p ) , where α ( X , p ) = max { | A | : A ∈ p } . This generalizes Hiltonʹs theorem on cross-intersecting families of finite sets, and provides analogs for cross-t-intersecting families of finite sets, finite vector spaces and permutations, etc. Moreover, the primitivity of symmetric systems is introduced to characterize the optimal families.