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 ⊂ A) and A ∩ Xi = B ∩ Xi for some i, then |F| ≤ ( n ⌊n/2⌋ ) . We consider variations of this problem by replacing the Sperner property with the intersection property and considering families that satisfiy 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 families of subsets of an n-element set such that F and G are both intersecting and cross-Sperner, meaning that if A ∈ F and B ∈ G, then A 6⊂ B and B 6⊂ A. Then |F|+ |G| < 2n−1 and there are exponentially many examples showing that this bound is tight.