Optimality of Wouter van Doorn's Upper Bound for the Mayer-Erdős Farey Problem

Let $\mathcal{F}_n$ be the Farey sequence of order $n$, written in increasing order. Call two fractions $\frac{a}{b} < \frac{c}{d}$ badly ordered if $a < c$ and $b > d$. Let $f(n)$ be the minimum number of Farey fractions strictly between two badly ordered fractions in $\mathcal{F}_n$. We prove $f(n)=\left(\frac{1}{4}+o(1)\right)n$. In the equivalent indexing convention of Erd\H{o}s Problem 1005, this determines the requested asymptotic constant as $c=1/4$. The upper bound $f(n)\le n/4+O(1)$ was first obtained by Wouter van Doorn; the main result here is the matching lower bound.