On the number of union-free families

TL;DR

This paper proves a conjecture about the number of union-free families of sets by developing a new container theorem for rooted hypergraphs, refining bounds on their quantity.

Contribution

It introduces a novel container theorem for rooted hypergraphs, resolving a longstanding conjecture on the enumeration of union-free families.

Findings

Confirmed the conjecture that the exponential constant can be removed

Established a new container theorem for rooted hypergraphs

Refined bounds on the number of union-free families

Abstract

A family of sets is union-free if there are no three distinct sets in the family such that the union of two of the sets is equal to the third set. Kleitman proved that every union-free family has size at most $(1+o(1))\binom{n}{n/2}$. Later, Burosch--Demetrovics-Katona-Kleitman-Sapozhenko asked for the number $\alpha(n)$ of such families, and they proved that $2^{\binom{n}{n/2}}\leq \alpha(n) \leq 2^{2\sqrt{2}\binom{n}{n/2}(1+o(1))}$. They conjectured that the constant $2\sqrt{2}$ can be removed in the exponent of the right hand side. We prove their conjecture by formulating a new container-type theorem for rooted hypergraphs.