How to find an integer set, s.t. the sums of at most 3 elements are all distinct? How to find a set $A \subset \mathbb{N}$ such that any sum of at most three Elements $a_i \in A$ is different if at least one element in the sum is different.
Example with $|A|=3$: Out of the set $A := \{1,7,11\}$ follow 19 sums 1,2,3,7,8,9,11,12,13,14,15,18,19,21,22,23,25,29,33 which are all distinct!
Is there an algorithm to construct such sets for a given number of elements $n=|A|$ with the smallest maximum possible?
 A: There is an algorithm: look at all $n$-element subsets of $S:=\{1,2,3,\dots,4^{n-1}\}$. There is at least one $B_3[1]$ subset among them, namely $\{1,4,4^2,\dots,4^{n-1}\}$. So list all $n$-element $B_3[1]$ subsets of $S$, and pick the one with the smallest maximum.
A: Note that there is a bit of a discrepancy between $B_3[1]$ sets and your notion.  For $B_3[1]$ sets, one considers sums with exactly 3 elements, instead of at most 3 elements.  However, I do not think this will make a big difference.  Basically the main point is that we can do much better than GH from MO's answer by taking a random subset of integers and using the probabilistic method.  Indeed, this was carried out by Erdős–Rényi for Sidon sets and claimed (but not proven) for $B_3[1]$ sets.  The first correct proof for $B_3[1]$ sets is attributed to Vu in this thesis of Carlos Vinuesa del Río.  That thesis also has the current state of the art for your problem.  See Theorem 2.1.2.  Specializing that result we have the following theorem.
Theorem. For any $\epsilon > 0$, there is an infinite $B_3[1]$ sequence $A$ such that $A(x) \gg x^{\frac{1}{5} - \epsilon}$. 
Here $A(x)$ is the number of elements of $A$ that are at most $x$.  
A: The probabilistic method shows the existence of an infinite $B_h[1]$ sequence of positive integers $A$ with $A(x)>x^{1/(2h-1)+o(1)}$. The proof is easy for $h=2$ but involved for $h\ge 3$. In particular it gives the exponent $1/5$ for $h=3$.
Ruzsa improved the exponent $1/3$ to $\sqrt 2-1$ for $h=2$ and
very recently Cilleruelo, using a new method, improved the exponent $1/(2h-1)$  to $\sqrt{(h-1)^2+1}-(h-1)$ for any $h\ge 3$. 
http://www.sciencedirect.com/science/article/pii/S000187081400022X
Furthermore, for $h=2$ this method provides an explicit $B_2[1]$ sequence $A$ with $A(x)>x^{\sqrt 2-1+o(1)}$.
