| Summary: | The union-closed sets conjecture states that, in any nonempty union-closed family <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">F</mi></semantics></math></inline-formula> of subsets of a finite set, there exists an element contained in at least a proportion <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></semantics></math></inline-formula> of the sets of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">F</mi></semantics></math></inline-formula>. Using an information-theoretic method, Gilmer recently showed that there exists an element contained in at least a proportion <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0.01</mn></mrow></semantics></math></inline-formula> of the sets of such <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">F</mi></semantics></math></inline-formula>. He conjectured that their technique can be pushed to the constant <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfrac><mrow><mn>3</mn><mo>−</mo><msqrt><mn>5</mn></msqrt></mrow><mn>2</mn></mfrac></semantics></math></inline-formula> which was subsequently confirmed by several researchers including Sawin. Furthermore, Sawin also showed that Gilmer’s technique can be improved to obtain a bound better than <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfrac><mrow><mn>3</mn><mo>−</mo><msqrt><mn>5</mn></msqrt></mrow><mn>2</mn></mfrac></semantics></math></inline-formula> but this new bound was not explicitly given by Sawin. This paper further improves Gilmer’s technique to derive new bounds in the optimization form for the union-closed sets conjecture. These bounds include Sawin’s improvement as a special case. By providing cardinality bounds on auxiliary random variables, we make Sawin’s improvement computable and then evaluate it numerically, which yields a bound approximately <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0.38234</mn></mrow></semantics></math></inline-formula>, slightly better than <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mfrac><mrow><mn>3</mn><mo>−</mo><msqrt><mn>5</mn></msqrt></mrow><mn>2</mn></mfrac><mo>≈</mo><mn>0.38197</mn></mrow></semantics></math></inline-formula>.
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