The Union-Closed Sets Conjecture for Small Families

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 $\mathcal{A}$.