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 field 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 $d{2}^{d-1}$? Recently, Kupavskii and Weltge [9] 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 $F_1,\dots ,F_{\ell }$ be families of subsets of $\{1,\dots ,n\}$. We suppose that for distinct $k,k^{\prime }$ and arbitrary $F_1\in F_k,F_2\in F_{k^{\prime }}$ we have $|F_1\cap F_2|⩽m$. We are interested in the maximal value of $\left|F_1\right|\dots \left|F_{\ell }\right|$ and the structure of the extremal example.

In the previous paper on the topic, the authors found the asymptotics of this product for constant ℓ 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.