# $B_k[1]$ sets with smallest possible $m = max B_k[1]$ for given $k$ and $n = |B_k[1]|$ elements

Sidon sets are sets $A \subset \mathbb{N}$ such that for all $a_j,b_j \in A$ holds $$a_1+a_2=b_1+b_2 \iff \{a_1,a_2\}=\{b_1,b_2\}$$ Thus if you know the sum of two elements, you know which elements were added in the first place.

The general case is known in literature as $B_k[g]$ sets. Every result of adding up at most k elements will occur at most g times in all possible combinations. Sidon sets are therefore called $B_2[1]$ sets.

An example of a $B_3[1]$ set

is $\{1,7,11\}$ with $k = 3$, $n = |B_k[1]|=3$ elements, $m = \max B_k[1] = 11$ and ${n+k \choose k}={6 \choose 3}=20$ distinct sums:

0=0, 1=1, 1+1=2, 1+1+1=3, 7=7, 7+1=8, 7+1+1=9, 11=11, 11+1=12, 11+1+1=13, 7+7=14, 7+7+1=15, 11+7=18, 11+7+1=19, 7+7+7=21, 11+11=22, 11+11+1=23, 7+7+11=25, 11+11+7=29, 11+11+11=33.

I didn't find a $B_3[1]$ set with $n=3$ and a smaller $m$ than 11. However $\{1, 8, 11\}$ has similar properties.

My Question

Considering $B_k[1]$ sets (only one solution for any sum of at most k elements). What is the smallest $m$ for given $k$ and $n$ and how to construct a set with this properties?

If the general case for any given $k$ isn't known, we may set $k=3$.

Note: In a former question ("Set of small numbers with distinct $k$-sums") a construction by Javier Cilleruelo was given. However it doesn't manage to find sets with the smallest $m$ possible.

• You might consider choosing a more informative title in order to make more people see this question. – Stefan Kohl Oct 16 '15 at 12:41