Improved Lower Bound for Frankl’s Union-Closed Sets Conjecture - Alweiss, Ryan; Huang, Brice; Sellke, Mark (The Electronic journal of combinatorics, 2024)
We verify an explicit inequality conjectured in [Gilmer, 2022, arXiv:2211.09055], thus proving that for any nonempty union-closed family $\mathcal{F} \subseteq 2^{[n]}$, some $i\in [n]$ is contained in at least a $\frac{3-\sqrt{5}}{2} ≈ 0.38$ fraction of the sets in $\mathcal{F} \$. One case, an explicit one-variable inequality, is checked by computer calculation.