Union-Closed Sets Conjecture: Improved Bounds

A union-closed family \(\mathcal{A} = \{A_1, A_2, \ldots, A_n\}\) is a non-empty finite collection of distinct non-empty finite sets, closed under union. It has been conjectured that for any such family, there is some element in at least half of its sets. But the problem remains unsolved. This paper establishes several results pertaining to this conjecture, with the main emphasis on the study of a possible counterexample with minimal \(n\).