Forbidding intersection patterns between layers of the cube

A family $\mathcal{A}\subset \mathcal{P}[n]$ is said to be an antichain if $A\subset ̸B$ for all distinct $A,B\in \mathcal{A}$. A classic result of Sperner shows that such families satisfy $|{\mathcal A}| \leq \binom {n}{\lfloor n/2\rfloor}$, 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 $\mathcal{P}[n]$. 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, we show that for most collections of such restrictions, layered families are asymptotically largest. This extends results of Leader and the author.