On a question of Erdos and Ulam

Ulam asked in 1945 if there is an everywhere dense \emph{rational set}, i.e. a point set in the plane with all its pairwise distances rational. Erd\H os conjectured that if a set $S$ has a dense rational subset, then $S$ should be very special. The only known types of examples of sets with dense (or even just infinite) rational subsets are lines and circles. In this paper we prove Erd\H os's conjecture for algebraic curves, by showing that no irreducible algebraic curve other than a line or a circle contains an infinite rational set.