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 ?

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 $m$, if $|\mathcal{G}| \geq 2^{m}$ then the average size, let's denote it $w(\mathcal{G})$, of a member of $\mathcal{G}$ is at least $m/2$. 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] D. Reimer, An average set size theorem, Combin. Probab. Computing 12 (2003), 89-93.