Erdős–Anning theorem

The theorem cannot be strengthened to give a finite bound on the number of points: there exist arbitrarily large finite sets of points that are not on a line and have integer distances.

The theorem is named after Paul Erdős and Norman H. Anning, who published a proof of it in 1945.

[1] Erdős later supplied a simpler proof, which can also be used to check whether a point set forms an Erdős–Diophantine graph, an inextensible system of integer points with integer distances.

The Erdős–Anning theorem inspired the Erdős–Ulam problem on the existence of dense point sets with rational distances.

Although there can be no infinite non-collinear set of points with integer distances, there are infinite non-collinear sets of points whose distances are rational numbers.

[2] For instance, the subset of points on a unit circle obtained as the even multiples of one of the acute angles of an integer-sided right triangle (such as the triangle with side lengths 3, 4, and 5) has this property.

[5] For any finite set S of points at rational distances from each other, it is possible to find a similar set of points at integer distances from each other, by expanding S by a factor of the least common denominator of the distances in S. By expanding in this way a finite subset of the unit circle construction, one can construct arbitrarily large finite sets of non-collinear points with integer distances from each other.

[1] Shortly after the original publication of the Erdős–Anning theorem, Erdős provided the following simpler proof.

[6][7] The proof assumes a given set of points with integer distances, not all on a line.

In more detail, it consists of the following steps: The same proof shows that, when the diameter of a set of points with integer distances is

[9] As Anning and Erdős wrote in their original paper on this theorem, "by a similar argument we can show that we cannot have infinitely many points in

The integer number line , a set of infinitely many points with integer distances. According to the Erdős–Anning theorem, any such set lies on a line.
The integer multiples of the angle of a 3–4–5 right triangle . All pairwise distances among the even multiples (every other point from this set) are rational numbers. Scaling any finite subset of these points by the least common denominator of their distances produces an arbitrarily large finite set of points at integer distances from each other.
Illustration for a proof of the Erdős–Anning theorem. Given three non-collinear points A , B , C with integer distances from each other (here, the vertices of a 3–4–5 right triangle), the points whose distances to A and B differ by an integer lie on a system of hyperbolas and degenerate hyperbolas (blue), and symmetrically the points whose distances to B and C differ by an integer lie on another system of hyperbolas (red). Any point with integer distance to all three of A , B , C lies on a crossing of a blue and a red curve. There are finitely many crossings, so finitely many additional points in the set. Each branch of a hyperbola is labeled by the integer difference of distances that is invariant for the points on that branch.
Five-vertex Erdős–Diophantine graph [ 9 ]