Forbidding intersection patterns between layers of the cube

A family A ⊂ P[n] is said to be an antichain if A 6⊂ B for all distinct A,B ∈ A. A classic result of Sperner shows that such families satisfy |A| ≤ ( n bn/2c ) , which is easily seen to be best possible. One can view the antichain condition as a restriction on the intersection sizes between sets in different layers of P[n]. More generally one can ask, given a collection of intersection restrictions between the layers, how large can families respecting these restrictions be? Answering a question of Kalai [8], we show that for most collections of such restrictions, layered families are asymptotically largest. This extends results of Leader and the author from [11].