An improved lower bound for the union-closed set conjecture

Gilmer has recently shown 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 $.01$ of the sets of $\mathcal{F}$. We improve the proportion from $.01$ to $\frac{3-\sqrt{5}}{2}\approx .38$ in this result. An improvement to $\frac{1}{2}$ would be the Frankl union-closed set conjecture. We follow Gilmer’s method, replacing one key estimate by a sharp estimate. We then suggest a new addition to this method and sketch a proof that it can obtain a constant strictly greater than $\frac{3-\sqrt{5}}{2}$. We also disprove a conjecture of Gilmer that would have implied the union-closed set conjecture.