Erdős Problem #21 — Smallest intersecting $n$-set family evading all $(n-1)$-covers

Problem of Lovász and myself, [32]. Let $f(n)$ be the smallest integer with the following property: There is a family $A_k$, $1 \leqq k \leqq f(n)$ satisfying $|A_k| = n$, $k = 1, \ldots, f(n)$, $|A_{k_1} \cap A_{k_2}| \geqq 1$ for every $1 \leqq k_1 < k_2 \leqq f(n)$, and for every $|S| = n-1$ there is an $A_k$ with $A_k \cap S = \emptyset$. In other words our family can not be represented by fewer than $n$ elements. We proved $f(n) < n^{3/2+\varepsilon}$. An improvement of our method very likely will give $f(n) < cn \log n$. I offer 500 dollars for a proof or disproof of $f(n) \leqq Cn$. In fact we can not even prove or disprove $f(n) < 3n$.