Dimension-Free Bounds for the Union-Closed Sets Conjecture
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The union-closed sets conjecture states that in any nonempty union-closed family ℱ of subsets of a finite set, there exists an element contained in at least a proportion 1/2 of the sets of ℱ. Using the information-theoretic method, Gilmer <cit.> recently showed that there exists an element contained in at least a proportion 0.01 of the sets of such ℱ. He conjectured that his technique can be pushed to the constant 3-√(5)/2≈0.38197 which was subsequently confirmed by several researchers <cit.>. Furthermore, Sawin <cit.> showed that Gilmer's technique can be improved to obtain a bound better than 3-√(5)/2. 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 around 0.38234, slightly better than 3-√(5)/2.
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