I asked this question on [Stackexchange](https://math.stackexchange.com/questions/1772398/union-closed-family-generated-by-n-2-sets), 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](https://mathoverflow.net/questions/75585/kruskal-katona-type-question-for-union-closed-families-of-sets)


but not identical.



Union-closed family generated by n 2-sets
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.](http://ajc.maths.uq.edu.au/pdf/52/ajc_v52_p067.pdf)", Australasian Journal of Combinatorics 52 (2012): 67-73.


