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Claim. Theorem 1 (p. 2): for all $n\ge4$, $f(n)\le\lfloor n/4\rfloor+d$ with $d=1,2,2,4$ for $n\equiv0,1,2,3\pmod4$, from explicit badly ordered pairs around $1/2$ (for $n=4m$, the fractions $(2m-1)/(4m)$ and $2m/(4m-1)$, at index distance $m+2$). Theorem 2 (p. 5): fractions of order $n$ at index distance at most $\frac n{12}(1-4n^{-1/3})$ are similarly ordered, so $f(n)\ge(\frac1{12}-o(1))n$, by optimizing Erdős's 1943 argument with a lemma on the arithmetic progressions formed by the Farey neighbors of a fraction of small denominator, and Dress's discrepancy bound. The Conjecture (p. 2): $f(n)>n/4$ for all $n\ge4$ and $f(n)=\lfloor n/4\rfloor+d$ for all $n\ge92$, checked for $n\le5000$, the exceptions below $92$ being $n=7,9,11,15,19,23,25,27,31,35,39,49,51,63,91$. The paper defines $f(n)$ as the site does (p. 1): the largest integer such that $a_k/b_k$ and $a_l/b_l$ are similarly ordered whenever $|l-k|\le f(n)$, so its theorems are statements about the $f(n)$ of [[problems/number_theory/E1005/_index|Problem 1005]] and need no bridge. The theorems are compiled on the result pages theorem_1 and theorem_2; the digest is on the card doorn_2025_improved_bounds_mayer_erdos_phenomenon_similarly.

Covers. The upper bound $f(n)\le\frac n4+O(1)$, with the explicit $d$, which is the upper half of the asymptotic $f(n)=(\frac14+o(1))n$, together with the lower bound $(\frac1{12}-o(1))n$. The matching lower bound $(\frac14-o(1))n$, which decides the problem's question, is on Cipollini's page; the exact-value conjecture for $n\ge92$ is not part of the problem and is claimed on the Wang--Xie--Zhao page.

Acceptance. Reviewed: the site's curator, Thomas F. Bloom, labels the problem solved and credits van Doorn, in the problem's commentary, with the bounds $(\frac1{12}-o(1))n\le f(n)\le\frac n4+O(1)$ and the conjecture that the upper bound is optimal, the resolution being these bounds together with Cipollini's lower bound; the curator neither wrote nor submitted the result. Not refereed: arXiv:2509.00121v1 (28 August 2025) is the only version, and no journal record was found (Crossref, 2026-09-18); the bounds are also the formula lines of OEIS A386893, van Doorn's own entry. The proofs were read for structure only, and nothing here is independently reviewed by this project.

Depends on. Nothing on the wiki; the theorems are proved in the preprint linked above.