Octopuses in the Boolean cube: families with pairwise small intersections, part I

Let $\mathcal F_1, \ldots, \mathcal F_\ell$ be families of subsets of $\{1, \ldots, n\}$. Suppose that for distinct $k, k'$ and arbitrary $F_1 \in \mathcal F_{k}, F_2 \in \mathcal F_{k'}$ we have $|F_1 \cap F_2|\le m.$ What is the maximal value of $|\mathcal F_1|\ldots |\mathcal F_\ell|$? In this work we find the asymptotic of this product as $n$ tends to infinity for constant $\ell$ and~$m$. This question is related to a conjecture of Bohn et al. that arose in the 2-level polytope theory and asked for the largest product of the number of facets and vertices in a two-level polytope. This conjecture was recently resolved by Weltge and the first author. The main result can be rephrased in terms of colorings. We give an asymptotic answer to the following question. Given an edge coloring of a complete $m$-uniform hypergraph into $\ell$ colors, what is the maximum of $\prod M_i$, where $M_i$ is the number of monochromatic cliques in $i$-th color?