Two-part set systems

The two part Sperner theorem of Katona and Kleitman states that if X is an n-element set with partition X₁ ∪ X₂, and F is a family of subsets of X such that no two sets(A, B)∈ F satisfy A ⊂ B (or B ε A) and A ∩ X_i = B ∩ X_i for some i, then F ≤ n. We consider variations of this problem by replacing the Sperner ⌊n/2⌋ property with the intersection property and considering families that satisfy various combinations of these properties on one or both parts X₁, X₂. Along the way, we prove the following new result which may be of independent interest: let F, G be intersecting families of subsets of an n-element set that are additionally cross-Sperner, meaning that if A ∈ F and B ∈ G, then A ⊂ B and B ⊂ A. Then |F| + |G| ≤ 2^n-1 and there are exponentially many examples showing that this bound is tight.