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Gilmer has recently shown that in any nonempty union-closed family $\mathcalF$ of subsets of a finite set, there exists an element contained in at least aproportion $.01$ 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 followGilmer's method, replacing one key estimate by a sharp estimate. We thensuggest a new addition to this method and sketch a proof that it can obtain aconstant strictly greater than $\frac{ 3 -\sqrt{5}}{2} $. We also disprove aconjecture of Gilmer that would have implied the union-closed set conjecture.

An improved lower bound for the union-closed set conjecture