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

TL;DR

This paper investigates the maximum product of sizes of families of subsets with pairwise small intersections, linking the problem to hypergraph colorings and a conjecture in polytope theory, providing asymptotic results for fixed parameters.

Contribution

It establishes the asymptotic maximum of the product of family sizes with small pairwise intersections, connecting combinatorial set systems to hypergraph colorings and polytope conjectures.

Findings

Asymptotic formula for the maximum product as n tends to infinity

Connection between set families with small intersections and hypergraph colorings

Resolution of a conjecture related to 2-level polytopes

Abstract

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?