The Union-Closed Sets Conjecture for Small Families

TL;DR

This paper proves the union-closed sets conjecture for small separating families with a size bound related to the number of elements, advancing understanding of the conjecture in specific cases.

Contribution

It establishes the conjecture's validity for certain small families with a specific size bound, providing new evidence for the conjecture.

Findings

Proves the conjecture for families with size up to a specific bound

Defines conditions under which the conjecture holds for separating families

Advances the understanding of the union-closed sets conjecture in small cases

Abstract

We prove that the union-closed sets conjecture is true for separating union-closed families $\mathcal{A}$ with $|\mathcal{A}| \leq 2\left(m+\frac{m}{\log_2(m)-\log_2\log_2(m)}\right)$ where $m$ denotes the number of elements in $\mathcal{A}$.