Problem 1005

Erdős Problem #1005: declared status 'open'. Formalized: no. Let $\frac{a_1}{b_1},\frac{a_2}{b_2},\ldots$ be the Farey fractions of order $n\geq 4$. Let $f(n)$ be the largest integer such that if $1\leq k<l\leq k+f(n)$ then $\frac{a_k}{b_k}$ and $\frac{a_l}{b_l}$ are similarly ordered - in other words,$$(a_k-a_l)(b_k-b_l)\geq 0.$$Estimate $f(n)$ - in particular, is there a constant $c>0$ such that $f(n)=(c+o(1))n$ for all large $n$? Current best: van Doorn [vD25b] has proved that$$\left(\frac{1}{12}-o(1)\right)n\leq f(n) \leq \frac{1}{4}n+O(1),$$and conjectures that the upper bound is optimal. Prize: no. OEIS: A386893. Tags: number theory.