TITLE: A certain kind of set of integers
QUESTION [7 upvotes]: Let us call a set $S$ of integers sparse if for any four integers $a,b,c,d \in S$ such that $a<b<c<d$, we have that $a+d \ne b+c$. Define $P(n)$ to be maximal size of a sparse subset of $\{1,2,\dotsc,n\}$. Is it true that $\lim\limits_{n\to \infty} P(n)/n=0$?

REPLY [4 votes]: Here's an explicit, tighter bound for the size of a sparse set.  If $A\subseteq \{1,2,\ldots,n\}$ is sparse, then every pair of distinct elements of $A$ has a distinct sum.  There are $\Theta(|A|^2)$ pairs of distinct elements, but only $O(n)$ different sums that they can have.  By the pigeonhole principle we conclude that $|A|$ is $O(\sqrt{n})$, and hence $|A|/n$ is $O(1/\sqrt{n})$.