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A family ${\mathcal A} \subset {\mathcal P} [n]$ is said to be an antichainif $A \not \subset B$ for all distinct $A,B \in {\mathcal A}$. A classic resultof Sperner shows that such families satisfy $|{\mathcal A}| \leq \binom{n}{\lfloor n/2\rfloor}$, which is easily seen to be best possible. One canview the antichain condition as a restriction on the intersection sizes betweensets in different layers of ${\mathcal P} [n]$. More generally one can ask,given a collection of intersection restrictions between the layers, how largecan families respecting these restrictions be? Answering a question of Kalai,we show that for most collections of such restrictions, layered families areasymptotically largest. This extends results of Leader and the author.

Forbidding intersection patterns between layers of the cube