Forbidding intersection patterns between layers of the cube

Abstract

A family $\mathcal{A}\subset \mathcal{P}[n]$ is said to be an antichain if $A\not\subset B$ for all distinct $A,B\in \mathcal{A}$. A classic result of Sperner shows that such families satisfy $|\mathcal{A}|\le \left(\genfrac{}{}{0ex}{}{n}{\lfloor n/2\rfloor }\right)$, 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.

16 pages

References in corpus (1)

• Frankl-Rödl type theorems for codes and permutations