Erdős Problem #20 — Exponential bound for families with no $k$-set sunflower

A family of sets $A_i$, $i = 1, 2, \ldots,$ is called a strong $\Delta$-system if all the intersections $A_i \cap A_j$ ($i \neq j$) are identical, i.e., if $A_i \cap A_j = \bigcap_i A_i$. Rado and I [40, 41] investigated the following question: Denote by $f_s(n,k)$ the smallest integer for which every family of sets $A_i$ ($1 \leq i \leq f_s(n,k)$) with $|A_i| = n$ for all $i$ contains $k$ sets which form a strong $\Delta$-system. Rado and I conjectured

$$f_s(n,3) < c_3^n$$

and no doubt also

$$f_s(n,k) < c_k^n.$$

I offer 1000 dollars for a proof or disproof of (11).