Question: Let $n,k$ be two positive integers with $n \geq k$. Let $\mathcal{F}$ be a family of $C(n,k)$ sets, each of size $k$, and let $\langle\mathcal{F}\rangle$ denote the union-closed family generated by $\mathcal{F}$, i.e.: $\langle\mathcal{F}\rangle$ consists of all those sets which can be expressed as a union of members of $\mathcal{F}$.
Must it be the case that \begin{equation} |\langle\mathcal{F}\rangle| \geq \sum_{j=k}^{n} C(n,j), \end{equation} with equality if and only if $\mathcal{F}$ consists of all $k$-element subsets of an $n$-set ?
Let $w(\mathcal G)$ denote the average size of the members of a family $\mathcal G$.
It is easy to see that if the inequality holds (whatever about uniqueness), then it implies that, for any union-closed family $\mathcal{G}$ and non-negative integer $m$, $$|\mathcal{G}| \geq 2^{m}\implies w(\mathcal{G})\ge m/2.$$ This is, in turn, a special case of a result of Reimer [1] that, for any union-closed family $\mathcal{G}$ one has $$w(\mathcal{G}) \geq \frac{1}{2} \log_{2} |\mathcal{G}|.$$ Indeed I had conjectured the same result and in thinking about it was led to the above question, before I recently became aware of Reimer's proof, which is a beautiful piece of work !
One can obviously try to generalise my question to an arbitrary number of generating $k$-sets, perhaps along the lines of the Kruskal-Katona theorem for shadows ?
[1] Reimer, David, An average set size theorem, Comb. Probab. Comput. 12, No. 1, 89-93 (2003). ZBL1013.05083.