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Octopuses in the Boolean cube: Families with pairwise small intersections, part II
The problem we consider originally arises from 2-level polytope theory. This class of polytopes generalizes a number of other polytope families. One of the important questions in this filed can be formulated as follows: is it true for a $d$-dimensional 2-level polytope that the product of the number of its vertices and the number of its $d-1$ dimensional facets is bounded by $d2^{d - 1}$? Recently, Kupavskii and Weltge~\cite{Kupavskii2020} settled this question in positive. A key element in their proof is a more general result for families of vectors in $\R^d$ such that the scalar product between any two vectors from different families is either $0$ or $1$.
Peter Frankl noted that, when restricted to the Boolean cube, the solution boils down to an elegant application of the Harris--Kleitman correlation inequality. Meanwhile, this problem becomes much more sophisticated when we consider several families.
Let $\ff_1, \ldots, \ff_\ell$ be families of subsets of $\{1, \ldots, n\}$. We suppose that for distinct $k, k'$ and arbitrary $F_1 \in \ff_{k}, F_2 \in \ff_{k'}$ we have $|F_1 \cap F_2|\le m.$ We are interested in the maximal value of $|\ff_1|\ldots |\ff_\ell|$ and the structure of the extremal example.
In the previous paper on the topic, the authors found the asymptotics of this product for constant $\ell$ and~$m$ as $n$ tends to infinity. However, the possible structure of the families from the extremal example turned out to be very complicated. In this paper, we obtain a strong structural result for the extremal families.