Farey sequence and Grahams conjectures

Let ${F}_{n}$ be the Farey sequence of order $n$. For $S subseteq {F}_n$ we let $mathcal{Q}(S) = left{x/y:x,yin S, xle y , , textrm{and} , , y eq 0right}$. We show that if $mathcal{Q}(S)subseteq F_n$, then $|S|leq n+1$. Moreover, we prove that in any of the following cases: (1) $mathcal{Q}(S)=F_n$; (2) $mathcal{Q}(S)subseteq F_n$ and $|S|=n+1$, we must have $S = left{0,1,frac{1}{2},ldots,frac{1}{n}right}$ or $S=left{0,1,frac{1}{n},ldots,frac{n-1}{n}right}$ except for $n=4$, where we have an additional set ${0, 1, frac{1}{2}, frac{1}{3}, frac{2}{3}}$ for the second case. Our results are based on Grahams GCD conjectures, which have been proved by Balasubramanian and Soundararajan.