Erdos-Anning Theorem

The Erdos-Anning Theorem states that whenever an infinite number of points in the plane all have pairwise integer distances from one another, those points must lie on a single straight line, and the same conclusion holds in higher-dimensional Euclidean spaces. The result cannot be strengthened into a finite bound on how many non-collinear points can share this property, since arbitrarily large finite sets of non-collinear points with all-integer pairwise distances do exist. Named after Paul Erdos and Norman Anning, who published a proof in 1945, with Erdos later giving a simpler proof, the theorem inspired the later Erdos-Ulam problem on dense sets of points with rational pairwise distances, and it has been independently rediscovered by other researchers, including with an application to the design of phased antenna arrays.