发表机构
Università degli Studi di Brescia; Government College of Engineering and Ceramic Technology, Kolkata（布雷西亚大学; 加尔各答政府工程与陶瓷技术学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文在 Liu 的框架内引入不同协议，结合独立协议与 Liu 的例 5，证明了并封闭集猜想的下界 $c_{\mathrm{UC}\ge c_{\mathrm L}$，并通过参数优化进一步将下界提升至 $0.3828852549667978$。
AI 中文摘要
并封闭集猜想询问是否每个包含非空集合的有限并封闭族都有一个元素至少出现在其一半的成员中。设 $c_{\mathrm{UC}}$ 表示该比例的最大通用下界。先前证明的最佳下界归功于 Liu，其值高于 $0.3823455$；他自己的数值优化建议了一个更强的阈值 $c_{\mathrm L}>0.382709087918735$。本文的主要论点是使用不同的协议（合适的概率测度族）在 Liu 的框架内。特别是，我们将独立协议与 Liu 的例 5 结合，证明了猜想的下界 $c_{\mathrm{UC}\ge c_{\mathrm L}$。然后我们改变例 5 中的参数，并优化组合中两个系数的选择，得到 $c_{\mathrm{UC}\ge 0.3828852549667978$。我们还证明了该证明中使用的不等式不能给出大于 $0.3828852599667\ldots$ 的阈值。
英文摘要
The union-closed sets conjecture asks whether every finite union-closed family containing a nonempty set has an element contained in at least half of its members. Let $c_{\mathrm{UC}}$ denote the largest universal lower bound on this proportion. The best previously proved lower bound is due to Liu and is above $0.3823455$; his own numerical optimization suggested a stronger threshold $c_{\mathrm L}>0.382709087918735$. The main argument of this paper is to use different protocols (suitable families of probability measures) within Liu's framework. In particular, we combine the independent protocol with Liu's Example 5 to prove the conjectured bound $c_{\mathrm{UC}}\ge c_{\mathrm L}$. We then vary the parameter in Example 5 and optimize the choice of the two coefficients in the combination to obtain $c_{\mathrm{UC}}\ge 0.3828852549667978$. We also show that the inequality used in this proof cannot give a threshold larger than $0.3828852599667\ldots$.
CommentsThe first version of this paper was made publicly available on Zenodo on September 7, 2026. The computational supplement and ancillary material are available at https://doi.org/10.5281/zenodo.22727713
