All Random Solved Random Open
SOLVED - $500
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 \[f(n) \ll n?\]

Conjectured by Erdős and Lovász [ErLo75], who proved that \[\frac{8}{3}n-3\leq f(n) \ll n^{3/2}\log n\] for all $n$. The upper bound was improved by Kahn [Ka92b] to \[f(n) \ll n\log n.\] (The upper bound constructions in both cases are formed by taking a random set of lines from a projective plane of order $n-1$, assuming $n-1$ is a prime power.)

This problem was solved by Kahn [Ka94] who proved the upper bound $f(n) \ll n$. The Erdős-Lovász lower bound of $\frac{8}{3}n-O(1)$ has not been improved, and it has been speculated (see e.g. [Ka94]) that the correct answer is $3n+O(1)$.

In [Er97f] Erdős asks about $f_\epsilon(n)$, defined analogously except with $\lvert S\rvert \leq n-1$ replaced by $\lvert S\rvert \leq (1-\epsilon)n$. He asks whether $f_\epsilon(n)/n\to \infty$ as $\epsilon \to 0$.

Additional thanks to: Zachary Chase