How small can "spanning sumsets" of $[n]$ be? Let $[n]$ denote the natural numbers $1$ through $n$. Let's say a subset $X \subset [n]$ is a spanning sumset if $\{x+y: x,y \in X\} = [n] \setminus \{1\}$. I'm interested in studying spanning sumsets of minimal possible size. In particular, is there either an asymptotic or exact expression for the minimal size of a spanning sumset? Is there an algorithm which can be used to find them? Or, in general, some sort of characterization? 
One simple observation is that $\{x+y: x,y \in X\}$ has at most $\frac{|X|(|X|+1)}{2}$ distinct sums, and if $\frac{|X|(|X|+1)}{2} \geq n-1$, then $|X| = \Omega(\sqrt{n})$. Can this lower bound be attained, however? That is to say, for general $n$ can we find a spanning sumset of size $\Theta(\sqrt{n})$? Or even just $o(n)$? 

Let $S_k=\{1,2,3,...,k,2k,3k,4k,...,(k-1)k\}$. 
Then $S_k$ is a spanning set for $[k^2]$.
For a given $n$ if we let $k=\lceil \sqrt{n} \rceil$, then $S_k$ will be a spanning set of $[n]$ with size $2k-2\approx 2\sqrt{n}$.
