Bulletin of the
Korean Mathematical Society BKMS

Let $F_n$ be the Farey sequence of order $n$. For $S\subseteq F_n$, let $\mathcal{Q}(S)$ be the set of rational numbers $x/y$ with $x,y\in S,~x\leq y$ and $y\neq 0$. Recently, Wang found all subsets $S$ of $F_n$ with $|S|=n+1$ for which $\mathcal{Q}(S)\subseteq F_n$. Motivated by this work, we try to determine the structure of $S\subseteq F_n$ such that $|S|=n$ and $\mathcal{Q}(S)\subseteq F_n$. In this paper, we determine all sets $S\subseteq F_n$ satisfying these conditions for $n\in\{p,2p\}$, where $p$ is prime.

Keywords: Farey sequences, Graham's conjectures

MSC numbers: Primary 11A05, 11B57

Supported by: This work was supported by the National Natural Science Foundations of China, Grant Nos. 12171243, 11901156 and 12201281, and the Natural Science Foundation of Youth of Henan Province, Grant No. 222300420245.