Let $\displaystyle m,n$ be positive integers and $\displaystyle A_1, A_2, ... , A_m \subseteq \left\{1,2,...,n\right\}$ such that $\displaystyle A_i \neq A_j$ for all $\displaystyle i \neq j$

Also, there exists a constant $\displaystyle L \in \mathbb{N}_0$ such that, forall$\displaystyle i \neq j$:

$\displaystyle \sum_{x \in A_i \cap A_j} x^3 = L$

Prove that $\displaystyle m \leq n$

...

It is obvious at a glance that there may only be one intersection of size one (if one exists), otherwise we get $\displaystyle L = a^3 = b^3$ but $\displaystyle a \neq b$. It's also easy to see that there has to be at least one intersection between some two sets.

Using Dilworth's theorem came to mind as we used it to prove some other stuff throughout the semester, however it didn't really yield any result, so I'm pretty much out of any ideas on how to proceed.

Any help would be appreciated!