Erdős–Anning theorem

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The Erdős–Anning theorem states that an infinite number of points in the plane can have mutual integer distances only if all the points lie on a straight line. It is named after Paul Erdős and Norman H. Anning, who proved it in 1945.

An alternative way of stating the theorem is that a non-collinear set of points in the plane with integer distances can only be extended by adding finitely many additional points, before no more points can be added. A set of points to which no more can be added, with all points on the integer grid, forms an Erdős-Diophantine graph.

[edit] Proof

Let A, B and C be non-collinear points with mutual distances D(AB), D(BC) and D(AC) not exceeding d, and X a point at integer distance from A, B and C. From the triangle inequality it follows that |D(AX) - D(BX)| is a non-negative integer not exceeding d. So X is on one of the d+1 hyperbolas through A and B. Similarly, X is situated on one of the d+1 hyperbolas through B and C. As two distinct hyperbolas can not intersect in more than four points, there are at most 4(d+1)² points X.