Here is a simple formulation for, I believe, a quite difficult problem.
I have played with it, I don't have the answer yet.
The question: How many pawns can you put on a standard 8x8 chess board in such a way that the distances between two pawns are all different?
Needless to say, each pawn must be exactly centered on a square of the board.
Computers are allowed. Without it, it is quite laborious to even check the validity of a solution.
If no proof of optimality is given (answering to "how many") then my vote goes to the solution that has the most pawns on the board.