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.


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.

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  • $\begingroup$ Thanks. My bet was : If $n={r\choose 2}+s$ with $s \leq r-1$ (which is posible in one and only one way), then $l(n) = 2^{r+1}-2^{r-s}-1-r$. Was I right ? (Sorry, stupid comment, since you gave a link.) $\endgroup$ – Panurge May 6 '16 at 11:18

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