Erdős–Ulam Problem

Statement:

Does the plane contain a dense set of points whose Euclidian distances are all rational numbers?

Author	Date Published
Paul Erdős and Stanislaw Ulam

Partial Results:

Tao [1] and Shaffaf [2] independently determined that the answer is no, provided the Bombieri-Lang Conjecture is true.

Pasten [3] showed that the abc Conjecture also implies the answer is no.

Related Problems:

Q808001 Bombieri–Lang conjecture

Q809001 Uniform Boundedness Conjecture for Rational Points

Q809002 Mazur's Conjecture B
References:

[1] Tao, Terence (2014-12-20), "The Erdos-Ulam problem, varieties of general type, and the Bombieri-Lang conjecture", What's new, retrieved 2016-12-05

[2] Shaffaf, Jafar (May 2018), "A solution of the Erdős–Ulam problem on rational distance sets assuming the Bombieri–Lang conjecture", Discrete & Computational Geometry, 60 (8), arXiv:1501.00159, doi:10.1007/s00454-018-0003-3

[3] Pasten, Hector (2017), "Definability of Frobenius orbits and a result on rational distance sets", Monatshefte für Mathematik, 182 (1): 99–126, doi:10.1007/s00605-016-0973-2, MR 3592123

Keywords:

Erdős–Ulam Problem; Euclidian distances; dense set;