Let $f(n)$ be the number of subsets $S\subseteq \{1,2,\ldots,2n\}$ such that $|S|=n$ and $a$ does not divide $b$ whenever $a,b \in S$ are distinct. Can we evaluate $f(n)$, at least asimptotically?
The question is related to this other one, where I had a more complicated (and unuseful) solution. In particular, is it true that $f(n)=o(n)$? Moreover, is it true that $f(n)$ is definitively bigger than $(\ln n)^k$ for any constant $k$?
Edit: the first conjecture has been proved to be false (see the answer of Robert below). Then, can we say that $\ln f(n)=O(n)$?
Here's one way to get a lower bound that grows exponentially. Start with the subset $S = \{n+1, \ldots, 2n\}$. Take an arbitrary subset $J \subseteq \{j \in S:\; j \text{ even},\; j > 4n/3\}$ and replace each $j \in J$ by $j/2$. This gives $2^{1 + \lfloor (n-1)/3 \rfloor}$ solutions.
EDIT: For the second question: of course $f(n)$ is at most the number of all subsets of $\{1, \ldots, 2n\}$, which is $2^{2n}$.