Erdős-Ulam Problem

The Erdős-Ulam problem asks whether there is a dense set of the real plane such that the Euclidean distance between every pair of its points is a rational number (Shaffaf 2018, Ascher et al. 2020). Density is required in the whole plane.

The problem remains open. It differs from finite rational distance problems, where denominators can be cleared to obtain integer distances. A single rescaling need not clear the denominators of infinitely many rational numbers.

This is problem 212 in the Erdős problems collection (Bloom 2026).