This is a problem that a professor proposed for the highschool mathematical olympiad in Costa Rica that we haven't been able to solve. Therefore it cannot be asked since we don't have a solution yet in general.
Let $\mathcal{F}_{k}$ for a fixed $k \in \mathbb{N}$ be the family of subsets $A_i \subset \mathbb{N}$ that satisfy the following conditions:
1) The cardinality of $A_i$ is $k$ for every index $i$.
2) For every $A_i$ it holds that given any two different subsets of two elements $ \{ x, y \} \neq \{ z, w \}$, the absolute value of the differences between the elements of each subset are different
$$ |x - y| \neq |z - w|$$
Now we define a function $f: \mathcal{F}_k \rightarrow \mathbb{N}$ given by $$f(A_i) = \max{A_i}$$
The problem is to find the minimum of the image of $f$, that is to find
$$\min{f(\mathcal{F}_k)}$$
For instance, we know that for $k = 4$ the answer is $\min{f(\mathcal{F}_4)} = 7$ and for $k = 3 $ the answer is $\min{f(\mathcal{F}_3)} = 4$ but we don't have a general pattern for the solution, basically these were done by "brute force".
We would very much appreciate any help with this problem. Thanks a lot in advance.