On a question of Erdos and Ulam

TL;DR

This paper proves Erdős's conjecture that only lines and circles contain infinite rational sets among algebraic curves, showing that other irreducible algebraic curves do not have such dense rational subsets.

Contribution

It establishes that no irreducible algebraic curve other than lines or circles contains an infinite rational set, confirming Erdős's conjecture for algebraic curves.

Findings

Only lines and circles contain infinite rational sets among algebraic curves

Erdős's conjecture holds for all irreducible algebraic curves

Other algebraic curves do not have dense rational subsets

Abstract

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.