Erdős Problem #1005

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 \(n\)? Mayer [Ma42] proved that \(f(n) \to \infty\) and Erdős [Er43] that \(f(n) \gg n\). Van Doorn [vD25b] proved \((1/12 - o(1)) n \le f(n) \le n / 4 + O(1)\) and conjectured that \(f(n) = (1/4 + o(1)) n\), which was proved by Cipollini and GPT-5.5; see `erdos_1005.variants.constant`.