Let $f(n)$ be minimal such that there is an intersecting family $\mathcal{F}$ of sets of size $n$ (so $A\cap B\ne \varnothing$ for all $A,B\in \mathcal{F}$) with $|\mathcal{F}|=f(n)$ such that any set $S$ with $|S|\le n-1$ is disjoint from at least one $A\in \mathcal{F}$.Is it true that$$f(n)\ll n?$$