TITLE: Union-closed family generated by n 2-sets
QUESTION [6 upvotes]: I asked this question on Stackexchange, but I got no answer, so I ask it here.
Let us define a $2$-set as a set with exactly $2$ elements. For a natural number $n$, let $l(n)$ denote the least possible number of members of a union-closed family of sets generated by $n$ distinct $2$-sets. I'm interested in a useful formula or minoration for $l(n)$.
There are easy majorations, for example $l({r\choose 2}) \leq 2^{r} - 1 - r$ and (for $r \geq 1$) $l({r\choose 2}+1) \leq 2^{r} + 2^{r-1} - 1 - r$, and these upper bounds seem to be exact values for small values of $r$, but I would avoid the task of handling this question if there is literature about it. Do you know ? Thanks in advance.
Note : there is a similar question here :
Kruskal-Katona type question for union-closed families of sets
but not identical.

REPLY [4 votes]: That problem was solved by Uwe Leck, Ian T. Roberts and Jamie Simpson in the paper "Minimizing the weight of the union-closure of families of two-sets.", Australasian Journal of Combinatorics 52 (2012): 67-73.