The Erdős–Ulam Problem

Erdős–Ulam problem: does there exist a dense subset \(S \subset \mathbb{R}^2\) such that the Euclidean distance between any two points of \(S\) is rational? (If \(S\) is dense and all mutual distances are rational, then after scaling, all mutual distances can be taken integral — so this asks for a dense integral point set in the plane.)

Stanisław Ulam asked it in 1946; Erdős popularised it, guessing no such set exists while emphasising that a proof seems far off. Solymosi–de Zeeuw (2010) ruled out the most natural candidates — no dense rational set lies on an irreducible algebraic curve — but the general question is untouched. Contrast the Anning–Erdős theorem (1945): an infinite set with all mutual distances integral must be collinear, so density forces the rational (not integral) formulation.