  Let A be a pre-defined set of rational numbers. We say a set of natural
numbers S is an A-quotient-free set if no ratio of two elements in S belongs to
A. We find the maximal asymptotic density and the maximal upper asymptotic
density of A-quotient-free sets when A belongs to a particular class. It is
known that in the case A = {p, q}, where p, q are coprime integers greater than
one, the latest problem is reduced to evaluation of the largest number of
lattice non-adjacent points in a triangle whose legs lie on coordinate axis. We
prove that this number is achieved by choosing points of the same color in the
checkerboard coloring.
