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A Note on the Union-closed Sets Conjecture
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abstract
Let $M$ be a non-zero binary matrix with distinct rows where the rows are closed under certain logical operators. In this article, we investigate the existence of columns containing an equal or greater number of ones than zeros. Specifically, the existence of such columns when the rows of the matrix are closed under $\textit{material conditional}$ leads us to a weaker version of the $\textit{Union-Closed Set Conjecture}$.
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Cited by 1 Pith paper
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On the Averaging Problem of Ideal Families Related to Frankl's Conjecture with Formal Proof by Lean 4
Every ideal family of sets is average rare: the average vertex degree is at most half the number of hyperedges, with a formal Lean 4 proof.
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