Math Conjectures

Erdős–Ulam Rational-Distance Problem

OPENMajorOpen problemProposed 1946 · Standard version

Canonical statement

Does there exist a countable dense set S⊂R2S\subset\mathbb R^2 such that
∥x−y∥2∈Qfor every x,y∈S? \|x-y\|_2\in\mathbb Q \qquad\text{for every }x,y\in S?
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Does there exist a countable dense set
\(S\subset\mathbb R^2\) such that
\[
  \|x-y\|_2\in\mathbb Q
  \qquad\text{for every }x,y\in S?
\]
DG-005:[Ulam1960Problems][BrassMoserPach2005][Solymosi2003RationalDistances]
Discrete geometryNumber theory

Notes

The Erdős–Ulam problem asks whether there is a countable set, dense in the plane, all of whose pairwise Euclidean distances are rational. The question is traditionally dated to 1946 and associated with both Erdős and Ulam; it appears in Ulam's problem collection [Ulam1960Problems] and is a staple of the discrete-geometry problem literature [BrassMoserPach2005].

Rational-distance sets can be large in constrained ways: there are configurations dense in curves, such as everywhere-dense rational-distance subsets of a line or of a circle. Addressing the question of Erdős and Ulam directly, Solymosi and de Zeeuw showed that infinite rational-distance sets contained in algebraic curves are essentially confined to lines and circles [Solymosi2003RationalDistances], which constrains the possible structure of any dense example.

No construction dense in the whole plane is known, and no impossibility theorem excludes one. The problem is open; either an explicit dense rational-distance set or a proof that density forces irrational distances would resolve it.

References (3)

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