For any positive integer $n$, let $X_n$ be the family of all subsets of $\{1,2,\cdots,n\}$.

Let $(X_n,d)$ be the metric space such that $$d(A,B)=|\,A\triangle B\,|,\ \forall A,B\in X_n$$ where $A\triangle B$ is the symmetric difference of $A$ and $B$.

Let real $k>0$ (be fixed), and let $\ S_n\subseteq X_n\ $ satisfy:

for every $F\in X_n$, there exists $E\in S_n$ such that $d(E,F)<k\sqrt{n}$.

Is

$$ \liminf_{n\rightarrow\infty}\frac{\mid S_n\mid}{\mid X_n\mid}>0\,? $$

Now I know the answer is no! I want to know more about quotients $\ \frac{|S_n|}{2^n}\ $ as $\ n\rightarrow\infty\, ?$