Dimension-Free Bounds for the Union-Closed Sets Conjecture

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 an information-theoretic method, Gilmer recently showed that there exists an element contained in at least a proportion 0.01 of the sets of such F. He conjectured that their technique can be pushed to the constant (3-root 5)/(2) 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 (3-root 5)/(2) 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 0.38234, slightly better than (3-root 5)/(2) approximate to 0.38197.