Abstract
A rational set in the plane is a point set with all its pairwise distances rational. Ulam asked in 1945 if there is an everywhere dense rational set. Solymosi and de Zeeuw proved that every rational distance subset of the plane has only finitely many points in common with an irreducible algebraic curve defined over R unless the curve is a line or circle. As an application of uniformity conjecture in arithmetic algebraic geometry which is a consequence of Lang conjecture we prove that if S is an infinite rational distance subset of the plane that has only finitely many points on any line then there is a uniform bound (independent of S) on the number of these points.
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