Abstract
We prove that the Union-Closed Sets Conjecture is true for separating union-closed families $$\mathcal {A}$$A with $$|\mathcal {A}| \le 2\left( m+\frac{m}{\log _2(m)-\log _2\log _2(m)}\right) $$|A|≤2m+mlog2(m)-log2log2(m) where m denotes the order of the universe $$\bigcup _{A\in \mathcal {A}} A$$źAźAA.
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