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\$.