Compute the maximum size of a 3-sunflower-free $n$-uniform family for small $n$ (Erdős #20)

A $k$-sunflower (or $\Delta$-system) is a collection of $k$ sets $A_1,\dots,A_k$ that pairwise share the same intersection $Y$ (the *core*): $A_i\cap A_j=Y$ for all $i\neq j$, equivalently the *petals* $A_i\setminus Y$ are pairwise disjoint. Let $f(n,k)$ be the maximum size of a family of $n$-element sets (an $n$-uniform family) that contains no $k$-sunflower. The Erdős–Rado sunflower conjecture asks whether, for each fixed $k$, $f(n,k)\le c_k^{\,n}$ for some constant $c_k>0$. Concrete target: determine the exact value of $f(n,3)$ (the largest $3$-sunflower-free $n$-uniform family) for small $n$, and exhibit extremal families. For example, $f(1,3)=2$; determine $f(2,3), f(3,3), f(4,3),\dots$ as far as computation allows.

Acceptance. FULLY RESOLVES (for a given $n$): the exact value of $f(n,3)$, i.e. (a) an explicit $3$-sunflower-free family of $n$-element sets of that size, plus (b) a machine-checkable certificate (exhaustive search / ILP / SAT) that no larger $3$-sunflower-free $n$-uniform family exists. PARTIAL: a new explicit large $3$-sunflower-free $n$-uniform family (improved lower bound) for a specific $n$, or an improved upper bound for specific $n$, each with a verification script that checks the sunflower-free property of the witness.