The union-closed sets conjecture states that in any nonempty union-closed family $\mathcal{F}$ of subsets of a finite set, there exists an element contained in at least a proportion $1/2$ of the sets of $\mathcal{F}$. Using the information-theoretic method, Gilmer \cite{gilmer2022constant} recently showed that there exists an element contained in at least a proportion $0.01$ of the sets of such $\mathcal{F}$. He conjectured that his technique can be pushed to the constant $\frac{3-\sqrt{5}}{2}$ which was subsequently confirmed by several researchers \cite{sawin2022improved,chase2022approximate,alweiss2022improved,pebody2022extension}. Furthermore, Sawin \cite{sawin2022improved} showed that Gilmer's technique can be improved to obtain a bound better than $\frac{3-\sqrt{5}}{2}$, but this new bound is 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 around $0.38234$, slightly better than $\frac{3-\sqrt{5}}{2}\approx0.38197$. }

翻译：并封闭集猜想指出：在有限集子集的任意非空并封闭族$\mathcal{F}$中，存在一个元素至少出现在$\mathcal{F}$中比例为$1/2$的集合中。Gilmer \cite{gilmer2022constant} 近期利用信息论方法证明，在这样的$\mathcal{F}$中至少存在一个元素出现在比例$0.01$以上的集合中。他推测其技术可推进至常数$\frac{3-\sqrt{5}}{2}$，这一结果随后被多位研究者证实\cite{sawin2022improved,chase2022approximate,alweiss2022improved,pebody2022extension}。此外，Sawin \cite{sawin2022improved} 表明Gilmer的技术可改进以获得优于$\frac{3-\sqrt{5}}{2}$的界，但该新界并未由Sawin显式给出。本文进一步改进Gilmer的技术，为并封闭集猜想推导出优化形式的新界——这些界将Sawin的改进作为特例包含在内。通过对辅助随机变量施加基数界，我们使Sawin的改进可计算，并对其数值评估得到约$0.38234$的界，略优于$\frac{3-\sqrt{5}}{2}\approx0.38197$。