Erdős–Anning theorem

Any set of points in the Euclidean plane whose pairwise distances are all integers is either finite or contained in a straight line.

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Integer distance problem: A geometric problem asking which point configurations have distances restricted to integers. The theorem is a central structural result about configurations with integer pairwise distances.
3–4–5 triangle: A right triangle with side lengths 3, 4, and 5. It is a simple noncollinear set whose three pairwise distances are integers.
Paul Erdős: A Hungarian mathematician known for prolific collaboration and work across combinatorics, number theory, and geometry. Erdős posed the integer-distance question that the theorem resolves.
Unit distance problem: The problem of determining how many pairs in a planar point set can be exactly one unit apart. It fixes one distance, whereas this theorem constrains every pairwise distance to be integral.
Maximum size of an integral point set: The problem of bounding or determining the largest finite planar set with integer pairwise distances under specified conditions. The theorem allows finite noncollinear sets but does not classify their possible sizes.
Distance geometry: The study of geometric configurations through the distances between their points. The theorem derives global shape restrictions from pairwise distance data alone.
Integer lattice: The set of points with integer coordinates in the Euclidean plane. Its points have integer pairwise distances only in restricted subsets, illustrating the theorem's constraint.
Richard Anning: A mathematician who co-authored the theorem on planar sets with integer pairwise distances. Anning established the result jointly with Erdős.
Erdős distinct distances problem: The problem of minimizing the number of distinct distances determined by a finite planar point set. It studies how many distances occur, rather than whether every distance is an integer.
Integral point set diameter: The greatest distance between points in a finite integral point set. Bounds on diameter help describe how large finite noncollinear examples can become.