The Mayer-Erdös phenomenon

Two rational numbers r and r′ are said to be similarly ordered if r = a/q, r′ = a′/q′, gcd(a, q) = gcd(a′, q′) = 1, and (a − a′) (q − q′) ⩾ 0. We investigate the problem of similar ordering of chains of consecutive Farey fractions.