# Progress on the union-closed sets conjecture

The union-closed sets conjecture is the following extremely easy to state conjecture about subsets of finite sets: Assume that $$\mathcal{F}$$ is a family of subsets of $$\{1, 2, \ldots, n\}$$ which is union-closed; this means that for any two sets $$A,B$$ in $$\mathcal{F}$$ their union $$A \cup B$$ is also a member of $$\mathcal{F}$$. Then there exists an element $$k \in \{1, 2, \ldots, n\}$$ which belongs to at least half of the members of $$\mathcal{F}$$.

In a recent preprint (only 9 pages long!) J. Gilmer now gave the first constant lower bound; concretely, he proved that there is an element $$k \in \{1, 2, \ldots, n\}$$ which belongs to at least a $$0.01$$ fraction of the sets in $$\mathcal{F}$$.

Only a few days later three other preprints (here, here, and here) were independently put on the arXiv improving Gilmer’s lower bound to $$(3-\sqrt{5})/2 \cong 0.38$$. Note that the conjecture postulates the lower bound of $$0.5$$.