Recently, Gilmer proved the first constant lower bound for the union-closed sets conjecture via an information-theoretic argument. The heart of the argument is an entropic inequality involving the OR function of two i.i.d.\ binary vectors, and the best constant obtainable through the i.i.d.\ coupling is $\frac{3-\sqrt{5}}{2}\approx0.38197$. Sawin demonstrated that the bound can be strictly improved by considering a convex combination of the i.i.d.\ coupling and the max-entropy coupling, and the best constant obtainable through this approach is around 0.38234, as evaluated by Yu and Cambie. In this work we show analytically that the bound can be further strictly improved by considering another class of coupling under which the two binary sequences are i.i.d.\ conditioned on an auxiliary random variable. We also provide a new class of bounds in terms of finite-dimensional optimization. For a basic instance from this class, analysis assisted with numerically solved 9-dimensional optimization suggests that the optimizer assumes a certain structure. Under numerically verified hypotheses, the lower bound for the union-closed sets conjecture can be improved to approximately 0.38271, a number that can be defined as the solution to an analytic equation.

翻译：近期，Gilmer通过信息论论证首次证明了并闭集猜想的常数下界。该论证的核心涉及两个独立同分布二进制向量的或函数熵不等式，通过独立同分布耦合可得最佳常数为$\frac{3-\sqrt{5}}{2}\approx0.38197$。Sawin指出，通过考虑独立同分布耦合与最大熵耦合的凸组合可严格改进该下界，Yu和Cambie评估该方法可得最佳常数约为0.38234。本文解析证明，通过考虑另一类耦合（两个二进制序列在辅助随机变量条件下独立同分布），可进一步严格改进该下界。我们还给出基于有限维优化的一类新下界。就此类基本实例而言，借助数值求解的9维优化分析表明，优化器呈现特定结构。在数值验证假设下，并闭集猜想的下降界可改进至约0.38271，该数值可定义为某个解析方程的解。