A family of sets $\mathcal{F} \subseteq 2^{[n]}$ is called union-closed if for any two sets $A,B \in \mathcal{F}$, $A \cup B \in \mathcal{F}$. We say that $\mathcal{F} \subseteq 2^{[n]}$ is sparse if the average size of a set in $\mathcal{F}$ is less than $n/2$. $\mathbb{E}_{A \in \mathcal{F}}|A| < n/2$.

Let's denote by $f(n)$ the number of sparse union-closed families of subsets of $[n]$. Any good bounds on $f(n)$?


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